# 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

**Question:** Find the nth term of this sequence 4,7,12,19,28?

**Answer:** First, work out the first differences; these are 3, 5, 7, 9.

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 3.

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

**Question:** What is the nth term of this quadratic sequence: 4,7,12,19,28?

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

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

Subtracting n^2 from the sequence gives 3, 3, 3, 3, 3.

So putting these two terms together gives n^2 + 3.

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

**Answer:** 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

**Question:** Find the nth term of this quadratic sequence 3,11,25,45?

**Answer:** The first differences are 8, 14, 20.

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 0,-1,-2,-3 which has the nth term of -n + 1.

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

**Question:** Find the nth term of 3,8,15,24?

**Answer:** The first differences are 5, 7, 9, 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 2, 4, 6, 8 which has nth term of 2n.

So putting both terms together gives n^2 + 2n.

**Question:** Can you find nth term of this quadratic sequence 2,8,18,32,50?

**Answer:** This is just the square number sequence doubles.

So if the square numbers have nth term of n^2, then the nth term of this sequence is 2n^2.

**Question:** Find the nth term of this sequence 6, 12, 20, 30, 42, 56, 72?

**Answer:** 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 5, 8, 11, 14, 17, 20, 23 which has nth term 3n + 2.

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

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

**Answer:** 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.

**Question:** Find the first three terms of this 3n+2?

**Answer:** You can find the terms by substituting 1,2 and 3 into this formula.

This gives 5,8,11.

**Question:** Find the nth term of this sequence 4,13,28,49,76?

**Answer:** The first differences of this sequence are 9, 15, 21, 27, and the second differences are 6.

Since half of 6 is 3 then the first term of the quadratic sequence is 3n^2.

Subtracting 3n^2 from the sequence gives 1 for each term.

So the final nth term is 3n^2 + 1.

**Question:** What is the nth term of this sequence:12, 17, 24, 33, 44, 57, 72?

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

This means that the first term of the sequence is n^2.

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

So putting these together gives a nth term of the quadratic sequence of n^2 + 2n + 9.

**Question:** What is the nth term of 3,8,17,30,47?

**Answer:** The first differences are 5, 9, 13, 17, and so the second differences are all 4.

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

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

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

**Question:** What is the Nth term of 4,9,16,25,36 ?

**Answer:** These are the square numbers, excluding the first term of 1.

Therefore, the sequence has a Nth term of (n+1)^2.

**Question:** Find the nth term of this sequence 3,8,15,24,35?

**Answer:** The first differences are 5, 7, 9, 11, and so the second differences are all 2.

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

Subtracting n^2 from the sequences gives 2,4,6,8,10 which has the nth term 2n.

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

**Question:** Find the nth term of this sequence 7, 14, 23, 34, 47, 62, 79 ?

**Answer:** The first differences are 7,9,11,13,15,17 and the second differences are 2.

This means that the first term of the sequence is n^2.

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

So putting these together gives a nth term of the quadratic sequence of n^2 + 4n + 2.

**Question:** What is the nth term of 6, 9, 14, 21, 30, 41?

**Answer:** These numbers are 5 more than the square number sequence 1,4,9,16,25,36 which has nth term n^2.

So the final answer for the nth term of this quadratic sequence is n^2 + 5.

**Question:** Find nth term of this sequence 4,11,22,37?

**Answer:** The first differences are 7, 11, 15, and the second differences are 4.

Since half of 4 is 2, then the first term will be 2n^2.

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

Therefore the final answer is 2n^2 + n + 1.

**Question:** Can you find the nth term of this sequence 8, 14, 22, 32, 44, 58, 74?

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

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

Subtracting n^2 from the sequence gives 7, 10, 13, 15, 18, 21, and the nth term of this linear sequence is 3n + 4.

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

**Question:** Find the nth term of this sequence 7,10,15,22,31?

**Answer:** These numbers are 6 more than the square numbers, so the nth term is n^2 + 6.

**Question:** What is the Nth term of 2, 6, 12, 20?

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

This means the first term is n^2.

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

So the final answer is n^2 + n.

**Question:** Find the nth term for 7,9,13,19,27 ?

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

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

Subtracting n^2 from the sequence gives 6,5,4,3,2 which has nth term -n + 7.

So the final answer is n^2 - n + 7.

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

**Answer:** 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.

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

**Answer:** 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.

**Question:** Find the sequence for n^2-3n+2?

**Answer:** 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.

**Question:** Can you find the nth term of this sequence 8,16,26,38,52,68,86?

**Answer:** 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.

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

**Answer:** 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.

**Question:** Find the nth term of this sequence 6, 10, 18, 30?

**Answer:** The first differences are 4, 8, 12, and so the second differences are all 4.

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

Subtracting 2n^2 from the sequences gives 4,2,0,-2, which has the nth term -2n + 6.

Therefore, the formula for this sequence is 2n^2 - 2n + 6.

**Question:** What is the nth term of this sequence 1,5,11,19?

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

This means the first term is n^2.

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

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

**Question:** Find the nth term of this sequence 2,8,18,32,50?

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

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

Subtracting 2n^2 from the sequence gives 0.

So the formula is just 2n^2.

**Question:** Write an expression in terms of n for 19,15,11?

**Answer:** This sequence is linear and not quadratic.

The sequence is going down by 4 each times so the nth term will be -4n + 23.

**Question:** If the nth term of a number sequence is n squared -3 what are the 1st, 2nd, 3rd and 10th terms?

**Answer:** The first term is 1^2 - 3 which is -2.

The second term is 2^2 -3 which is 1

The third term is 3^2 -3 which is 6.

The tenth term is 10^2 - 3 which is 97.

**Question:** Find the nth term for this sequence -5,-2,3,10,19?

**Answer:** The numbers in this sequence are 6 less than the square numbers 1, 4, 9, 16, 25.

Therefore the nth term is n^2 - 6.

**Question:** Find the nth term of this number sequence 5,11,19,29?

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

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

Subtracting n^2 from this sequence gives 4, 7, 10, 13 which has nth term 3n + 1.

So the final nth term formula is n^2 + 3n + 1.

**Question:** Can you find the nth term of 4,7,12..?

**Answer:** These numbers are three more than the square number sequence 1,4,9, so the nth term will be n^2 + 3.

**Question:** Can you find the nth term 11,14,19,26,35,46?

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

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

**Answer:** 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.

**Question:** Find the nth term of this quadratic sequence 2 7 14 23 34 47?

**Answer:** The first differences are 5, 7, 9, 11, 13, and the second differences are 2.

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

Subtracting n^2 gives 1, 3, 5, 7, 9, 11 which has nth term 2n - 1.

Therefore the nth term is n^2 + 2n - 1.

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

**Answer:** 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).

**Question:** Can you find the nth term for this quadratic sequence 1,2,4,7,11?

**Answer:** 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.

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

**Answer:** 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).

**Question:** How can I find the next terms of this sequence 4,16,36,64,100?

**Answer:** 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.

**Question:** What is the nth term of 7,10,15,22,31,42?

**Answer:** 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.

**Question:** Find nth term of this sequence 4,10,18,28,40?

**Answer:** The first differences are 6, 8,10,14 and the second differences are 2.

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

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

Therefore, the final nth term is n^2 + 3n.

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

**Answer:** 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.

**Question:** Can you find the nth term of 11, 26, 45 and 68?

**Answer:** The first differences are 15, 19 and 23. The second differences are 4.

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

Subtracting 2n^2 from the sequence gives you 9, 18, 27 and 36, which has nth term 9n.

So, the final formula for this quadratic sequence is 2n^2 + 9n.

**Question:** What is the nth term rule of this quadratic sequence: 8, 14, 22, 32, 44, 58, 74?

**Answer:** The first differences are 6, 8, 10, 12, 14, 16, and so the second differences are all 2.

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

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

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

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

**Answer:** 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.

**Question:** What is the nth term rule of the quadratic sentence? -7,-4,3,14,29,48

**Answer:** 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.

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

**Answer:** 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.

**Question:** What is the nth term rule of the sequence -8, -8, -6, -2, 4?

**Answer:** 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.

## Comments

**hellom8** on February 14, 2019:

this is very helpful , cheers

**confusedchild** on September 05, 2018:

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

**louisearabiana** on June 18, 2018:

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

**Mark (author)** from England, UK on November 24, 2017:

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

**its ya boi** on November 19, 2017:

What if the second difference is an odd number?

**Silver49** on January 05, 2014:

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

**MKING** on September 12, 2013:

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** on June 17, 2013:

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** on May 09, 2013:

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** on May 09, 2013:

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** on April 23, 2013:

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

quad eqns made easy for us math phobs people.

**shafster** on November 30, 2012:

its epic

**sid** on May 17, 2012:

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

**Angel** on February 12, 2012:

N is number of term.

**Angel** on February 12, 2012:

Dats cul

**matthias caruana** on November 27, 2011:

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

it should be 11,15,19,23

**Me.** on April 04, 2011:

thanks for this, had trouble understanding for my maths homework