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Quadratic Sequences: The Nth Term of a Quadratic Number Sequence

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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

1st difference = 11,15,19,23

2nd 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...

1st differences = 19,29,39,49

2nd 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