# Quadratic Sequences: The Nth Term of a Quadratic Number Sequence

Here, we will be finding the nth term of a quadratic number sequence. A quadratic number sequence has nth term = an² + bn + c

**Example 1 **

Write down the nth term of this quadratic number sequence.

-3, 8, 23, 42, 65...

**Step 1:** Confirm the sequence is quadratic. This is done by finding the second difference.

**Sequence **= -3, 8, 23, 42, 65

**1 ^{st} difference** = 11,15,19,23

**2 ^{nd} difference** = 4,4,4,4

**Step 2:** If you divide the second difference by 2, you will get the value of a.

4 ÷ 2 = 2

So the first term of the nth term is 2n²

**Step 3:** Next, substitute the number 1 to 5 into 2n².

**n **= 1,2,3,4,5

**2n²** = 2,8,18,32,50

**Step 4:** Now, take these values (2n²) from the numbers in the original number sequence and work out the nth term of these numbers that form a linear sequence.

**n **= 1,2,3,4,5

**2n²** = 2,8,18,32,50

**Differences ** = -5,0,5,10,15

Now the nth term of these differences (-5,0,5,10,15) is 5n -10.

So b = 5 and c = -10.

**Step 5**: Write down your final answer in the form an² + bn + c.

2n² + 5n -10

**Example 2**

Write down the nth term of this quadratic number sequence.

9, 28, 57, 96, 145...

**Step 1:** Confirm if the sequence is quadratic. This is done by finding the second difference.

**Sequence** = 9, 28, 57, 96, 145...

**1 ^{st} differences** = 19,29,39,49

**2 ^{nd} differences** = 10,10,10

**Step 2:** If you divide the second difference by 2, you will get the value of a.

10 ÷ 2 = 5

So the first term of the nth term is 5n²

**Step 3:** Next, substitute the number 1 to 5 into 5n².

**n **= 1,2,3,4,5

**5n²** = 5,20,45,80,125

**Step 4:** Now, take these values (5n²) from the numbers in the original number sequence and work out the nth term of these numbers that form a linear sequence.

**n = **1,2,3,4,5

**5n² = **5,20,45,80,125

**Differences = **4,8,12,16,20

Now the nth term of these differences (4,8,12,16,20) is 4n. So b = 4 and c = 0.

**Step 5:** Write down your final answer in the form an² + bn + c.

5n² + 4n

## Questions & Answers

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Find nth term of this sequence 10,33,64,103?

The first differences are 23, 31, 39 and the second difference is 8.

Therefore since half of 8 is 4 the first term will be 4n^2.

Subtracting 4n^2 from the sequence gives 6, 17, 28 which has nth term 11n - 5.

So the final answer is 4n^2 + 11n -5.

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Find the nth term of this sequence 2,9,20,35,54?

The first differences are 7, 11, 15, 19.

The second differences are 4.

Half of 4 is 2, so the first term of the sequence is 2n^2.

If you subtract 2n^2 from the sequence you get 0,1,2,3,4 which has the nth term of n - 1

Therefore your final answer will be 2n^2 + n - 1

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What is the ninth term of this sequence 6,12,20,30,42,56?

The first differences are 6,8,10,12,14. The second difference is 2. Therefore half of 2 is 1 so the first term is n^2. Subtract this from the sequence gives 5,8,11,14,17. The nth term of this sequence is 3n + 2. So the final formula for this sequence is n^2 + 3n + 2.

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Find the nth term of this sequence 8,14, 22, 32, 44, 58, 74?

The first differences are 6,8,10,12,14,16, and the second differences are 2.

Half of 2 is 1, so the first term is n^2.

Subtracting n^2 from the sequence is 7, 10, 13, 16, 19, 22, 25 which has nth term 3n +4.

So the final answer is n^2 + 3n + 4.

Find the sequence for n^2-3n+2?

First sub in n = 1 to give 0.

Next sub in n =2 to give 0.

Next sub in n = 3 to give 2.

Next sub in n = 4 to give 6.

Next sub in n = 5 to give 12.

Keep going to find other terms in the sequence.

Can you find the nth term of this sequence -3,0,5,12,21,32?

The first differences are 3,5,7,9,11, and the second differences are 2.

Therefore the first term in the quadratic sequence is n^2.

Subtracting n^2 from the sequence gives -4.

So the final answer of this sequence is n^2 -4.

(Just subtract 4 from your square number sequence).

Helpful 3Can you find the nth term for this quadratic sequence 1,2,4,7,11?

The fist differences are 1, 2, 3, 4 and the second difference are 1.

Since the second differences are 1, then the first term of the nth term is 0.5n^2 (Half of 1).

Subtracting 0.5n^2 from the sequence gives 0.5,0,-0.5,-1,-1.5 which has nth term -0.5n + 1.

So the final answer is 0.5n^2 - 0.5n + 1.

Helpful 1What is the nth term of 7,10,15,22,31,42?

The first differences are 3,5,7,9,11 and the second differences are 2.

The first term of the sequence is, therefore, n^2 (since half of 2 is 1).

Subtracting n^2 from the sequence gives 6.

So putting these 2 terms together gives a final answer of n^2 + 6.

Helpful 3Can you find the nth term 11,14,19,26,35,46?

This sequence is 10 higher than the square number sequence, so the formula is nth term = n^2 + 10.

Helpful 1What is the nth term rule of the quadratic sequence below? − 8 , − 8 , − 6 , − 2 , 4 , 12 , 22...?

The first differences are 0, 2, 4, 6, 8, 10.

The second differences are 2.

Half of 2 is 1, so the first term of the sequence is n^2.

If you subtract n^2 from the sequence gives -9, -12, -15, -18, -21, -24, -27 which has nth term -3n - 6.

Therefore your final answer will be n^2 -3n - 6.

Helpful 1What is the nth term of this fractional number sequence 1/2 , 4/3, 9/4, 16/5?

First look for the nth term of the numerators of each fraction (1,4,9,16). Since these are square numbers then the nth term of this sequence is n^2.

The denominators of each fraction are 2,3,4,5, and this is a linear sequence with nth term n + 1.

So putting these together the nth term of this fractional number sequence is n^2/(n+1).

Helpful 3How can I find the next terms of this sequence 4,16,36,64,100?

These are the even square numbers.

2 squared is 4.

4 squared is 16.

6 squared is 36.

8 squared is 64.

10 squared is 100.

So the next term in the sequence will be 12 squared which is 144, then the next one 14 squared which 196 etc.

Helpful 3Can you find the nth term of this sequence 8,16,26,38,52,68,86?

The first differences are 8,10,12,14,16,18 and the second differences are 2.

Since half of 2 is 1, then the first term of the nth term is n^2.

Subtracting n^2 from the sequence gives 7,12,17,22,27,32,37 which has a nth term of 5n + 2.

So putting these together gives a nth term of the quadratic sequence of n^2 + 5n + 2.

Helpful 5What is the nth term rule of the quadratic sequence below? − 5, − 4, − 1, 4, 11, 20, 31, . . .

The first differences are 1, 3, 5, 7, 9, 11, and the second differences are 2.

Half of 2 is 1 so the first term is n^2.

Take this from the sequence to give -6, -8, -10, -12, -14, -16, -18 which has nth term of -2n - 4.

So the final answer is n^2 - 2n - 4.

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What is the nth term of this: 3,18,41,72,111?

The first differences are 15,23,31,39, and the second differences are 8.

Halving 8 gives 4, so the first term of the formula is 4n^2

Now subtract 4n^2 from this sequence to give -1,2,5,8,11, and the nth term of this sequence is 3n – 4.

So the nth term of the quadratic sequence is 4n^2 + 3n – 4.

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What is the nth term of 6, 20, 40, 66, 98,136?

The first differences are 14, 20, 26, 32 and 38, and so the second differences are all 6.

Halving 6 gives 3, so the first term of the sequence is 3n^2.

Subtracting 3n^2 from the sequences gives 3,8,13,18,23 which has the nth term 5n-2.

Therefore, the formula for this sequence is 3n^2 + 5n - 2.

Helpful 9What is the nth term rule of the quadratic sentence? -7,-4,3,14,29,48

The first differences are 3,7,11,15,19 and the second differences are 4.

Halving 4 gives 2, so the first term of the formula is 2n^2.

Now subtract 2n^2 from this sequence to give -9,-12,-15,-18, -21, -24 and the nth term of this sequence is -3n -6.

So the nth term of the quadratic sequence is 2n^2 – 3n – 6.

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Can you find the nth term of this sequence 8,16,26,38,52?

The first difference of the sequence are 8, 10, 12, 24.

The second differences of the sequences are 2, therefore since half of 2 is 1 then the first term of the sequence is n^2.

Subtracting n^2 from the given sequence gives, 7,12,17,22,27. The nth term of this linear sequence is 5n + 2.

So if you put the three-term together, this quadratic sequence has the nth term n^2 + 5n + 2.

Helpful 7Can you find the nth term of this sequence 4, 7, 12, 19, 28?

These numbers are 3 more than the square numbers 1, 4, 9, 16, 25, so the nth term is n^2 + 3.

Helpful 1Can you find the nth term of the quadratic sequence 3,8,17,30,47?

First work out the first differences, these are 5, 9, 13, 17.

Next, find the second differences, these are all 4.

So since half of 4 is 2, then the first term is 2n^2.

Subtracting 2n^2 from the sequence gives 1, 0, -1, -2, -3 which has nth term of -n + 2.

So the nth term of this quadratic sequence is n^2 -n + 2.

Helpful 6What’s is the nth term of this: -4,1,12,29?

The first differences are -5, -11, -17, and so the second differences are all 6.

Halving 6 gives 3, so the first term of the sequence is 3n^2.

Subtracting 3n^2 from the sequences gives -7,-11,-15,-19 which has the nth term -4n-3.

Therefore, the formula for this sequence is 3n^2 -4n - 3.

Helpful 6Can you find the nth term of this sequence 2,5,10,17?

The first differences are 3, 5, 7 and the second difference are 2.

Since the second differences are 2, then half of 2 is 1, so the first tem of the sequence is n^2.

Subtracting n^2 from this squences gives 1,1,1,1.

So the nth term of this quadratic sequence is n^2 + 1.

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Find the nth term rule of this sequence 12,17,24,33,44,57,72?

The first differences are 5,7,9,11,13,15 and the second differences are 2.

Therefore the first term of the sequence is n^2.

Subtracting n^2 from the sequence gives 11, 13, 15, 17, 19, 21, 23 which has nth term 2n + 9.

So the formula is just n^2 + 2n + 9.

Helpful 2Find the nth term of this sequence 5,9,21,41?

The first differences are 4, 12, 20.

The second differences are 8.

Therefore, since 1/2 of 8 = 4, then the first term of the quadratic sequence is 4n^2.

Subtract 4n^2 from the sequence gives 1, -7, -15, -23 and the nth term of the linear sequence is -8n + 9.

Hence the nth term of this quadratic sequence is 4n^2 -8n + 9.

Helpful 5Can you find the nth term of this sequence 4,7,12,19,28?

The first difference are 3, 5, 7 and 9. The second differences 2, so the first term is n^2 (since half of 2 is 1).

Subtracting n^2 from the sequence gives 3.

So the nth term of this quadratic sequence is n^2 + 3.

Helpful 5Can you find the next term in the sequence 4,12,24,40?

The first differences are 8, 12, 16, so the next difference will be 20 as the first differences are increasing by 4. Therefore the next term is 40 + 20 = 60.

Helpful 5What is the nth term rule of the sequence -8, -8, -6, -2, 4?

The first differences are 0, 2, 4, 6, and the second diffferences are all 2.

Since half of 2 is 1, then the first term of the quadratic nth term is n^2.

Next, subtract n^2 from the sequence to give -9,-12,-15,-18,-21 which has nth term -3n - 6.

So the nth term will be n^2 -3n - 6.

Helpful 5Find the nth term of this sequence -1,2,7,14,23?

These numbers are 2 less than the square number sequence 1,4,9,16,25....

Therefore the nth term will be n^2 - 2

Helpful 1How would I find the nth term of this sequence 2,10, 24, 44?

The first differences are 8, 14, 20 and the second difference is 6.

Since half of 6 gives 3, the first term of the sequence is 3n^2.

Subtracting 3n^2 from this sequence gives -1, -2, -3, -4, which has a nth term of -n.

Therefore putting this together gives an answer of 3n^2 -n.

Helpful 4How would I find the nth term of this sequence: 4, 7, 12, 19, 28?

First find the first differences to give 3, 5, 7, 9 and therefore the second differences are 2.

This means the first term will be n^2 as half of 2 is 1.

Subtracting n^2 from the sequence gives 3.

So the final formula for the sequence is n^2 + 3.

Helpful 4Can you find the nth term of 2,17,40,71?

The first differences are 15, 23, 31, and so the second differences are all 8.

Halving 8 gives 4, so the first term of the sequence is 4n^2.

Subtracting 4n^2 from the sequences gives -2, 1, 4, 7 which has the nth term 3n - 5

Therefore, the formula for this sequence is 4n^2 + 3n - 5.

Helpful 4How can you work out the nth term for this quadratic sequence 6, 10, 16, 24, 34?

The first differences are 4,6,8,10, and the second differences are 2 giving the first term of n^2.

Subtracting n^2 from this sequence gives 5,6,7,8,9 which has nth term of n + 4.

So the nth term of this quadratic sequence is n^2 + n + 4.

Helpful 4Find the nth term of this 2, 17, 40, 71?

The fist differences are 15, 23, 31, and the second difference are 8.

Since the second difference are 8, then the first term of the nth term is 4n^2 (Half of 8).

Subtracting 4n^2 from the sequence gives -2, 1, 4, 7, which has nth term 3n -5.

So the final answer is 4n^2 + 3n -5.

Helpful 4Is the number 174 in the sequence n²+5?

Make n^2 + 5 = 174.

Subtract 5 from both sides to give n^2 = 169.

Then square root both sides to give n = 13.

174 is the 13th term in the sequence, so yes.

Helpful 4What are the next 3 terms of 26,37,50?

Assuming that the sequence is quadratic the first difference are 11 and 13. So the first differences are increasing by 2 each times.

So the fourth term will be 50 + 15 = 65.

The fifth term will be 65 + 17 = 82.

The sixth term will be 82 + 19 which is 101.

Helpful 4What is the nth term rule of the quadratic sequence: -5,-4,-1,4,11,20,31?

The first differences in the sequence are 1, 3, 5, 7, 9, 11 and the second differences are all 2.

Halving that equals 1, so the first term of the sequence is n^2.

Subtracting n^2 from the sequence gives -6, -8, -10, -12, -14, -16, -18, which has nth term -2n - 4.

So the final answer is n^2 -2n - 4.

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Find the third term of 5n^2?

First work out 3^2 to give 9.

Then multiply 9 by 5 to give 45.

(You just need to do the squaring before the multiplying).

Helpful 1What is the nth term of this sequence of numbers 0,1,4,9,16?

This is your square number sequence starting at 0 instead of 1.

Therefore the nth term will be (n-1)^2.

Helpful 3How would I find the nth term of this sequence 3,13, 27, 45, 67?

The first differences are 10, 14, 18, 22 and the second differences are 4. Therefore the first term is 2n^2 as half os 4 gives 2.

Subtracting 2n^2 from the sequence gives 1, 5, 9, 13, 17 which has nth term of 4n -3.

Putting all this together givers 2n^2 + 4n - 3.

Helpful 3What is the nth term of this: 4, 10, 18, 28, 40?

The first differences are 6,8,10,12, and the second differences are 2.

Since half of 2 is 1, then the first term of the nth term is n^2.

Subtracting n^2 from the sequence gives 3,6,9,12,15 which has a nth term of 3n.

So putting these together gives a nth term of the quadratic sequence of n^2 + 3n.

Helpful 3Find the nth term of this sequence 4, 7, 12, 19, 28?

These numbers are 3 more than the square number sequence, so the nth term is n^2 + 3.

Helpful 2What is the nth term of -4,-1,4,11,20,31?

The first differences are 3, 5, 7, 9, 11 and the second differences are all 2, so the sequence must be quadratic.

Half of 2 gives 1, so the first term of the nth term is n^2.

Subtracting n^2 from the sequence gives -5.

So putting both terms together gives n^2 -5.

Helpful 3What is the nth term of this quadratic sequence : -4,1,12,29,52,81?

The first differences are 5, 11, 17, 23, 29, and the second differences are 6.

Halving 6 gives 3, so the first term will be 3n^2.

Subtracting 3n^2 from the sequence gives -7, -11, -15, -19, -23, -27, which has nth term -4n -3.

So the final answer will be 3n^2 -4n - 3.

Helpful 3Find the nth term for the quadratic sequence of 3,9,17,27?

The first differences are 6 8,10 and the second differences are 2.

The first term of the sequence is therefore n^2 (since half of 2 is 1).

Subtracting n^2 from the sequecne gives 2, 5, 8, 11 which has the nth term of 3n - 1.

So putting the 3 terms together gives a final answer of n^2 + 3n - 1.

Helpful 3Find the nth term if this sequence 8,16,26,38,52,68,86?

The first differences are 8,10,12,14,16,18, and the second differences are 2, so the first term will be n^2.

Subtracting n^2 from the sequence gives 7,12,17,22,27,32,37 which has nth term of 5n + 2.

So the final answer is n^2 + 5n + 2.

Helpful 3What is the nth term of the quadratic sequence 4,7,12,19,28?

The first differences are 3, 5, 7, 9 and the second diffferences are all 2.

Since half of 2 is 1, then the first term of the quadratic nth term is n^2.

Next subtract n^2 from the sequence to give 3 for each term.

So the nth term will be n^2 + 3.

Helpful 3What is the nth term of this quadratic number sequence: 1,3,6,10,15?

The first differences are 2,3,4,5 and the second differences are 1.

Therefore the first term in the quadratic sequence is 0.5n^2.

Subtracting 0.5n^2 from the sequence gives 0.5, 1, 1.5, 2, 2.5, and the nth term of this linear sequence is 0.5n.

So the final answer of this sequence is 0.5n^2 + 0.5n.

Helpful 3Can you find the nth term of this quadratic sequence 4,7,12,19,28?

The first differences are 3, 5, 7 and 9, and the second differences are 2.

Therefore the first term will be n^2 (half of 2 is 1).

Subtracting n^2 from the sequences gives 3.

So the final answer is n^2 + 3.

Helpful 3Find the nth term of this sequence 9,12,17,24?

These numbers are 8 more than the square number sequence, so the nth term is n^2 + 8.

Helpful 3Find the nth term for this sequence 4, 9, 16, 25, 36?

(n+1)^2

Find nth term of this 0, 3, 8, 15, 24?

These numbers are 1 less than the square numbers, so the nth term is n^2 - 1.

Helpful 1Find the nth term of this sequence -1 2 7 14 23?

These numbers are two smaller than the square number sequence (1,4,9,16,25...)

Since the nth term of the square numbers is n^2 then the formula for this sequence will be n^2-2.

Helpful 1find the nth term of this quadratic sequence 4,7,12,19,28?

These numbers are 3 more than the square number sequence which is n^2.

So the final answer is n^2 + 3.

Helpful 1What is the sequence formula for 6,17,34,57,86?

The first differences are 11, 17, 23, 29.

The second differences are 6.

Half of 6 is 3, so the first term of the sequence is 3n^2.

If you subtract 3n^2 from the sequence you get 3,5,7,9 which has the nth term of 2n + 1.

Therefore your final answer will be 3n^2 + 2n + 1.

Helpful 2Can you find the nth term of this sequence -1, 5, 15, 29, 47?

The first differences are 6, 10, 14, 18, and the second differences are 4.

Half of 4 is 2, so the first term is 2n^2.

Take this from the sequence to gives -3 for each term.

So the final answer is 2n^2 - 3.

Helpful 1What is the nth term of this quadratic number sequence 5, 9, 1, 29?

The first differences are 4, 8, 12, and the second differences are 4.

Since half of 4 is 2 then the first term of the nth term is 2n^2.

Subtracting 2n^2 from this sequence gives 3, 1, -1, -3 which has nth term of -2n + 5.

So putting this together gives 2n^2 -2n + 5.

Helpful 2What is the nth term of the sequence 3, 13, 27, 45?

The first differences are 10, 14, 18, and the second differences are all 4, so the sequence must be quadratic.

Half of 4 gives 2, so the first term of the nth term is 2n^2.

Subtracting 2n^2 from the sequence gives 1, 5, 9, 13 which has nth term of 4n -3.

So the final nth term is 2n^2 + 4n - 3.

Helpful 2Find the nth term of the sequence 1,11,27,49?

The first differences are 10, 16, 22, and the second difference are 6 meaning that the sequence is quadratic and the first term being 3n^2.

Subtracting 3n^2 from the sequence gives -2,-1,0,1 which has nth term n - 3.

So putting these terms together gives a final answer of 3n^2 + n - 3.

Helpful 2Find the nth term of this sequence 4,16,36,64,100?

This sequence is 4 times bigger than the square number sequence, so the nth term is 4n^2.

Helpful 2What are the first three terms for n^(2)+20?

To work this out the numbers 1,2 and 3 need to be substituted into the nth term.

So the first term is 1^2 + 20 = 21.

The second term is 2^2 + 20 = 24.

The third term is 3^2 + 20 = 29.

So the first three terms are 21,24,29.

Helpful 2Can you find the nth term of this quadratic sequence. 3,11,25,45?

First, work out the first differences; these are 8, 14, 20.

Next, find the second differences, these are all 6.

So since half of 6 is 3, then the first term is 3n^2.

Subtracting 3n^2 from the sequence gives 0, -1, -2, -3 which has nth term of -n + 1.

So the nth term of this quadratic sequence is 3n^2 -n + 1.

Helpful 2How do I find the nth term of this sequence 0, 5, 12, 21, 32?

First differences are 5,7,9,11, and the second differences are 2.

The first term will, therefore, be n^2, as half of 2 is 1.

Subtracting n^2 from the sequence gives -1,1,3,5,7 which has nth term 2n - 3.

So the final nth term is n^2 + 2n - 3.

Helpful 2What is the nth term of this quadratic sequence 11,15, 23, 35?

The fist differences are 4, 8, 12, and the second difference are 4.

Since the second difference are 4, then the first term of the nth term is 2n^2 (Half 4).

Subtracting 2n^2 from the sequence gives 9, 7, 5, 3, which has nth term -2n + 11.

So the final answer is 2n^2 - 2n + 11.

Helpful 2Can you find the value of k of this sequence 1,-5,k,-23?

Assuming the sequence is quadratic then the first differences are -6, k + 5, and -23 - k.

The second differences are k + 11 and -2k and -28.

Since the second differences are the same of a quadratic sequence then k + 11 = -2k - 28.

This gives 3k = -39, and k = -13.

Helpful 2Find the nth term of this sequence 0,6,18,36,60?

First, work out the first differences; these are 6, 12, 18, 24.

Next, find the second differences, these are all 6.

So since half of 6 is 3, then the first term is 3n^2.

Subtracting 3n^2 from the sequence gives -3, -6, -9, -12, -15 which has nth term of -3n

So the nth term of this quadratic sequence is 3n^2 - 3n.

Helpful 2What is the nth term for this sequence 6, 9, 14, 21, 30?

The fist differences are 3, 5, 7, 9, and the second difference are 2.

Since the second differences are 2, then the first term of the nth term is n^2 (Half 2).

Subtracting n^2 from the sequence gives 5.

So the final answer is n^2 + 5.

Helpful 2Find the nth term of the sequence 1,7,17,31?

The first differences are 6, 10, 14.

The second differences are 4.

Half of 4 is 2, so the first term of the sequence is 2n^2.

If you subtract 2n^2 from the sequence you get -1 for each term.

Therefore your final answer will be 2n^2 - 1

Helpful 1Can you find the nth term of -7, -6, -3, 2, 9, 18, 29?

First differences are 1, 3, 5, 7,9, 11.

Second differences are 2.

First term is therefore n^2 (Since half of 2 is 1)

Subtracing n^2 from the sequence gives -8, -10, -12, -14, -16, -18, -20 which has nth term -2n - 6.

So the final answer is n^2 - 2n - 6.

What is the nth term rule of the quadratic sequence below? − 4 , − 3 , 0 , 5 , 12 , 21 , 32 , .

The first differences are 1, 3, 5, 7, 9, 11.

The second differences are 2.

Half of 2 is 1, so the first term of the sequence is n^2.

If you subtract n^2 from the sequence gives -5, -7, -9, -11, -13, -15, -17 which has nth term -2n - 3.

Therefore your final answer will be n^2 -2n - 3.

Helpful 1What is second and fifth term of the sequence 3n^2-1?

Substitute n = 2 to work out the second term, so 2^2 is 4, 4 times 3 is 12, and subtract 1 is 11.

Substitute n = 5 to work out the fifth term, so 5^2 is 25, 3 times 25 is 75, and subtract 1 is 74.

Can you find the nth term of 10,24,44,70,102,140?

First work out the first differences, these are 14, 20, 26, 32, 38

Next find the second differences, these are all 6.

So since half of 6 is 3, then the first term is 3n^2.

Subtracting 3n^2 from the sequence gives 7, 12, 17, 22, 27 which has nth term of 5n + 2.

So the nth term of this quadratic sequence is 3n^2 + 5n + 2.

Helpful 1What is the nth term of the sequence 4 6 10 16 24 34 46?

The first differences are 2,4,6,8,10,12, and the second differences are 2.

So the first term of the sequence is n^2.

Subtracting n^2 from the sequence gives 3,2,1,0,-1,-2,-3 which has nth term -n + 4.

So the nth term is n^2 - n + 4.

Helpful 1Can you find the nth term of this sequence 2,14,36,68?

First find the first differences to give 12, 22, 32, and therefore the second differences are 10.

This means the first term will be 5n^2 as half of 10 is 5.

Subtracting 5n^2 from the sequence gives -3, -6, -9, -12, which has nth term -3n.

So the final formula for the sequence is 5n^2 - 3n.

Helpful 1What is the Nth term of 8, 16, 28 ,44?

First find the first differences to give 8, 12, 16, and therefore the second differences are 4.

This means the first term will be 2n^2 as half of 4 is 2.

Subtracting 2n^2 from the sequence gives 6, 8, 10, 12, which has nth term of 2n + 4.

So the final formula for the sequence is 2n^2 + 2n + 4.

Helpful 1Can you find the nth term of this sequence 1, 6, 15, 28, 45?

The first differences are 5, 9, 13, 17, and the second differences are 4.

Half of 4 is 2, so the first term is 2n^2.

Take this from the sequence to give -1, -2, -3, -4, -5 which has a nth term of -n.

So the final answer is 2n^2 - n.

Find the nth term for this sequence : 4,7,12,19,28?

These terms in the sequence are 3 higher than the square number nth term, so the nth term is n^2 + 3

Helpful 1Can you find nth term of this sequence 9, 15, 23, 33, 45?

The first differences are 6,8,10,12, and the second differences are 2.

Therefore the first term in the sequence is n^2 as half of is 1.

Subtracting n^2 from the sequence gives 8,11,14,17,20 which has nth term 3n + 5.

So the final answer is n^2 + 3n + 5.

Helpful 1Can you find the nth term of this sequence 1, 3, 7, 13, 21?

The first differences are 2,4,6,8, and the second differences are 2.

Therefore, the first term will be n^2.

Subtracting n^2 from the sequence gives 0,-1,-2,-3,-4 which has nth term of -n + 1.

So the final answer is n^2 - n + 1.

Helpful 1What's the nth term of the sequence 1,3,9,19,33,51?

First differences are 2, 6,10,14,18.

Second differences are 4, so the first term is 2n^2.

Minus 2n^2 from the sequence to give, -1,-5-9,-13,17 which has nth term -4n + 3.

So the final answer is 2n^2 -4n + 3.

Helpful 1What is the nth term of this quadratic number sequence 3,7,12,18?

The first differences are 4,5,6, and the second differences are 1, so the first term will be 0.5n^2.

Subtracting 0.5n^2 from the sequence gives 2.5,5,7.5,10 which has nth term of 2.5n.

So the final answer is 0.5n^2 + 2.5n.

Helpful 1What is the nth term of this sequence -8,-2,8,22,40?

The fist differences are 6, 10, 14, 18 and the second difference are 4.

Since the second differences are 4, then the first term of the nth term is 2n^2 (Half 4).

Subtracting 2n^2 from the sequence gives -10.

So the final answer is 2n^2 - 10.

Helpful 1Find the nth term of this sequence 4,7,12,19,28,39?

The first differences are 3, 5, 7, 9, 11 and the second differences are 2.

Half of 2 is 1, so the first term is n^2.

Subtract n^2 from the sequence gives 3.

So the nth term of the sequence is n^2 + 3.

Helpful 1What is the nth term of this sequence -5, -4, -1, 4, 11, 20, 31?

The first differences are 1, 3, 5, 7, 9, 11 and the second differences are 2.

Half of 2 is 1, so the first term is n^2.

Take this from the sequence to give -6, -8, -10, -12, -14, -16, -18 which has nth term of -2n-4.

So the final answer is 2n^2 - 2n - 4.

Find the nth term of this sequence 8,14,22,32,44,58,74?

First differences are 6, 8, 10, 12, 14, 16.

Second differences are 2.

First term is therefore n^2 (Since half of 2 is 1)

Subtracing n^2 from the sequence gives 7, 10, 13, 16, 19, 22, 25 which has nth term 3n + 4.

So the final answer is n^2 + 3n + 4.

Can you find the nth term of 3, 10, 21, 36, 55?

First differences are 7, 11, 15, 19.

Second differences are 4.

The first term is therefore 2n^2 (Since half of 4 is 2).

Subtracting 2n^2 from the sequence gives 1, 2, 3, 4, 5, which has nth term n.

So the final answer is 2n^2 + n.

Find the nth term of the sequence 302,600,894,1184?

The first differences are 298, 294, 290.

The second differences are -4.

Half of -4 is 2, so the first term of the sequence is -2n^2.

If you subtract n^2 from the sequence gives 304, 608, 912, 1216 which has nth term 304n.

Therefore your final answer will be -2n^2 + 304n.

What are the first 4 term of n squared +3?

Sub in n=1, 1^2 + 3 = 4.

Next sub in n=2, 2^2 + 3 = 7.

Next sub in n = 3, 3^2 + 3 = 12.

Next sub in n = 4, 4^2 + 3 = 19.

So the first 4 terms are 4,7,12,19.

How do I find the nth term of the quadratic sequence -1, 2, 7, 14, 23?

These numbers are two less than the square numbers, so the nth term is n^2 - 2.

what is the nth term of:4,9,16,25,36 ?

The first differences are 5, 7, 9, 11.

The second differences are 2.

Half of 2 is 1, so the first term of the sequence is n^2.

If you subtract n^2 from the sequence you get 3,5,7,9,11 which has the nth term of 2n + 1.

Therefore your final answer will be n^2 + 2n + 1.

Find the nth term of this sequence 4,9,14,19?

This sequence is linear not quadratic as the first differences are the same.

It has nth term 5n -1.

Find the nth term of this sequence 5,5,3,-1,-7?

The first differences are 0, -2, -4, -6, and the second differences are all 2.

Half of -2 is -1 so the first term is -n^2.

Subtracting -n^2 from the sequence gives 6, 9, 12, 15, 18 which has nth term 3n + 3.

So the final nth term of this quadratic sequecne is -n^2 + 3n + 3.

How do you find the second term of this sequence 3, x, 29, 51?

The first differences are x - 3, 29 - x, and 21. The second differences are 32 - 2x and -8 + x.

Since the second differences are the same then 32 - 2x = -8 + x.

So 3x = 40, and x = 40/3 or 13 1/3.

Find the nth erm of this sequence 10,22,40,64,94,130?

The first differences are 12, 18, 24, 30, 36, and the second differences are 6.

This means the first term is 3n^2 (half of 6).

Subtracting 3n^2 from the sequence gives 7, 10, 13, 16, 19, 22 which has nth term 3n + 4.

So the nth term of this quadratic is 3n^2 + 3n + 4.

Could you please find the nth term of this sequence 3, 9, 17, 27, and 39?

The first differences are 6, 8, 10, 12.

The second differences are 2.

Half of 2 is 1, so the first term of the sequence is n^2.

If you subtract n^2 from the sequence you get 2,5,8,11,14 which has the nth term of 3n - 1

Therefore your final answer will be n^2 + 3n - 1

The nth term for 10, 13, 18, 25?

These numbers are 9 more than the square numbers, so the nth term is n^2 + 9.

What is the ninth term of this sequence? 2,11,26,47,74,107?

First work out the nth term.

The first differences are 9,15,21,27,33, and the second differences are 6.

Therefore the first term of the sequence is 3n^2.

Subtracting 3n^2 from the sequence gives -1.

So the formula is 3n^2 - 1.

Now substitute n = 9 into this formula, 3 multiplied by 9^2 - 1 gives 242.

Can you find the nth term of this sequence: 12,17,24,33,44,57,72?

First differences are 5,7,9,11,13,15

Second differences are 2, so the first term is n^2.

Minus n^2 from the sequence to give, 11,13,15,17,19,21,23 which has nth term 2n + 9.

So the final answer is n^2 + 2n + 9.

How do you find nth term of this quadratic sequence 3,6,11,18,27,38?

The numbers in the sequence are 2 higher than the the square number sequence of 1,4,9,16,25,36.

Therefore the nth term is n^2 + 2.

What is the nth term rule of the quadratic sequence below? 7,14,23,34,47,62,79

The first differences are 7,9,11,13,15,17,19,21 and the second difference are 2.

Half of 2 is 1, so the first term is n^2.

Subtract n^2 from the sequence gives 6, 10, 14, 18, 22, 26, 30 which has nth term 4n + 2.

So the nth term of the sequence is n^2 + 4n + 2.

Can you find the nth term rule of this pattern 8, 14, 22, 32, 44, 58, 74?

The first differences are 6, 8, 10, 12, 14, 16 and the second difference are 2.

Half of 2 is 1, so the first term is n^2.

Subtract n^2 from the sequence gives 7, 10, 13, 16, 19, 22, 25 which has nth term 3n + 4.

So the nth term of the sequence is n^2 + 3n + 4.

Find the expression of this sequence? -5,-2,5,16,31

The first differences are 3,7,11,15, and the second differences are 4.

Therefore the first term of the sequence is 2n^2.

Subtracting 2n^2 from the sequence gives -7, -10, -13, -16, -19 which has nth term -3n - 4.

So the formula is 2n^2 -3n - 4.

What is the nth term of this sequence: 3,18,41,72,111?

The first differences are 15,23,31,39, and the second differences are 8

Therefore the first term of the sequence is 4n^2.

Subtracting 4n^2 from the sequence gives -1, 2, 5, 8, 11 which has nth term 3n - 4.

So the formula is just 4n^2 + 3n - 4.

Find the nth term of this sequence 3 9 19 33 51?

The first differences are 6, 10, 14, 18.

The second differences are 4.

Half of 4 is 2, so the first term of the sequence is 2n^2.

If you subtract 2n^2 from the sequence you get 1 for each term.

Therefore your final answer will be 2n^2 + 1.

Find the nth term of this sequence -4, -3,0 , 5 , 12 , 21 ,32?

The first differences are 1, 3, 5, 7, 9, 11.

The second differences are 2.

Half of 2 is 1, so the first term of the sequence is n^2.

If you subtract n^2 from the sequence you get -5,-7, -9 , -11, -13. -15, -17 which has the nth term of -2n -3.

Therefore your final answer will be n^2 - 2n - 3.

Find the nth term of this sequence 2,17,40,71?

First, work out the first differences; these are 15, 23, 31.

Next, find the second differences, these are all 8.

So since half of 8 is 4, then the first term is 4n^2.

Subtracting 4n^2 from the sequence gives -2, 1, 4, 7 which has nth term of 3n - 5.

So the nth term of this quadratic sequence is 4n^2 +3n - 5

Find the nth term of this sequence 1,3,7,13,21?

First, work out the first differences; these are 2, 4, 6,8.

Next, find the second differences, these are all 2.

So since half of 2 is 1, then the first term is n^2.

Subtracting n^2 from the sequence gives 0, -1, -2, -3, -5 which has nth term of -n + 1.

So the nth term of this quadratic sequence is n^2 -n + 1.

Can you find the nth term of this sequence 5,16,33,56?

The first differences are 11, 17, 23, and the second differences are 6.

Half of 6 is 3, so the first term is 3n^2.

Subtracting 3n^2 from the sequence gives 2, 4, 6, 8 which has nth term 2n.

So the final formula for this quadratic sequence is 3n^2 + 2n.

Find the nth term of this sequence -2,1,6,13,22,33?

The numbers in this quadratic sequence are 3 smaller than the square number sequence, so the nth term is n^2 - 3.

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## Popular

## Comments

**hellom8**4 months ago

this is very helpful , cheers

**confusedchild**9 months ago

im still a bit confused how you work out 'c' like how did you get -10 / 0?

**louisearabiana**12 months ago

where does this difference came from, like the 4,8,12,16,20?

- AUTHOR
**Mark**19 months ago from England, UK

This would rarely happen, but you can still apply the same method. Just put a decimal before n squared.

**its ya boi**19 months ago

What if the second difference is an odd number?

**Silver49**5 years ago

Very good explanation I will recommend this site to any one who has got problem understanding quadratic sequence

**MKING**5 years ago

its it the same if you get for example 3n+3n+3n+1 like 3 nth terms do u just go and make a third sequence from your second sequence

**cool guy**6 years ago

what do you do if the number only repeats after 4 differences:

1 1 2 7 21

0 1 5 14 1

1 4 9 2

3 5 3

2 4

What is the equation?

**Gervasius Stephanus**6 years ago

Conceptual undertanding is very important for students to comprehend quadratic sequence. Hence, the logic of determining the terms of a given sequence defined by a quadratic formula should be the starting point.

**Dr Gervasius Stephen**6 years ago

The procedure helps far better but understanding of the concept per se in changing the sequence to arithmetic number sequence should be greatly emphasised.

**MathFabPhobiaCured**6 years ago

dat is just dam cul. luving dis site. helpd me a lot.

quad eqns made easy for us math phobs people.

**shafster**6 years ago

its epic

**sid**7 years ago

brilliant i was having trouble with quadratic sequences and this helped a lot thanks

**Angel**7 years ago

N is number of term.

**Angel**7 years ago

Dats cul

**matthias caruana**7 years ago

In step one there's a mistake in the first difference

it should be 11,15,19,23

**Me.**8 years ago

thanks for this, had trouble understanding for my maths homework

17