Quadratic and Cubic Sequences

Quadratic and cubic sequences: comparing with squares and cubes, finding which term a number is, and second differences.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 35 real past-paper questions below

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Look for the Square Numbers

If the gaps grow, the sequence is not linear. Compare it with the square numbers 1, 4, 9, 16, 25.

Worked example

1
$6, 9, 14, 21, 30$ — The sequence. The gaps are 3, 5, 7, 9. They grow, so it is not linear.
2
$1, 4, 9, 16, 25$ — Write the square numbers underneath: $1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}$ .
3
$6 = 1 + 5, 9 = 4 + 5$ — Compare each term with the square in the same place. Every term is 5 more.
4
$n th term = n^{2} + 5$ — So square the position n, then add 5.
5
$10 th term = 1 0^{2} + 5 = 105$ — Put n = 10. Square first: $1 0^{2} = 100$ , then add 5.

Common mistake

Adding 9 again and again from 30. The gaps grow each time, so a fixed step misses the next term.

Terms From the nth Term

Exams often give you the nth term and ask for the first few terms. Put n = 1, then 2, then 3.

Worked example

1
$n = 1 : 1^{2} - 7 = 1 - 7 = - 6$ — Put n = 1 into $n^{2} - 7$ . Square first, then take 7. The answer is negative, and that is fine.
2
$n = 2 : 2^{2} - 7 = 4 - 7 = - 3$ — Put n = 2. 4 − 7 goes below zero again, so the term is −3.
3
$n = 3 : 3^{2} - 7 = 9 - 7 = 2$ — Put n = 3. 9 − 7 = 2.
4
$- 6, - 3, 2$ — Write the terms in order. These are the first three terms.

A number in front of $n^{2}$ multiplies the square. It does not get squared.

Worked example

1
$n = 1 : 4 \times 1^{2} = 4$ — For $4 n^{2}$ , square n first: $1^{2} = 1$ . Then times 4.
2
$n = 2 : 4 \times 2^{2} = 4 \times 4 = 16$ — Square 2 to get 4, then times 4.
3
$n = 3 : 4 \times 3^{2} = 4 \times 9 = 36$ — Square 3 to get 9, then times 4. The terms are 4, 16, 36.

Common mistake

Working out $(4 n)^{2}$ and getting 16, 64, 144. In $4 n^{2}$ only the n is squared.

Common mistake

Starting at n = 0. The first term is always n = 1.

Which Term Is It?

To test if a number is in the sequence, set the nth term equal to it and solve for n.

Worked example

1
$n^{2} + 5 = 789$ — Is 789 a term of $n^{2} + 5$ ? Set the nth term equal to 789.
2
$n^{2} = 784$ — Take 5 from both sides, so $n^{2}$ is alone.
3
$n = 784 = 28$ — Square root both sides. n is a position, so only the positive root makes sense.
4
$789 is the 28 th term$ — n = 28 is a whole number, so 789 is in the sequence.

Key idea

If n is not a whole number, the number is not a term. Is 600 a term of $n^{2} + 5$ ? $n^{2} = 595$ and $595 = 24.39 \dots$ , so no.

Key idea

Show the equation you solved. It usually earns a mark even if the answer is wrong.

Harder Term Questions (Extended/Higher)

Two harder question types: a cubic nth term with negatives, and the smallest positive term.

Worked example

1
$50 - n^{3} = - 293$ — Which term of $50 - n^{3}$ is −293? Set the nth term equal to it.
2
$- n^{3} = - 293 - 50 = - 343$ — Take 50 from both sides. −293 − 50 = −343.
3
$n^{3} = 343$ — Multiply both sides by −1. Both minus signs go.
4
$n = 3343 = 7$ — Cube root: 7 × 7 × 7 = 343. So −293 is the 7th term.

Find the smallest positive term of $n^{2} - 200$ :

Worked example

1
$n^{2} - 200 > 0$ — Positive means bigger than 0.
2
$n^{2} > 200$ — Add 200 to both sides.
3
$n > 200 = 14.14 \dots$ — Square root. n is a position, so it is positive.
4
$n = 15$ — n must be a whole number. The first whole number above 14.14 is 15.
5
$1 5^{2} - 200 = 25$ — Put n = 15 into the nth term. The smallest positive term is 25.
6
$1 4^{2} - 200 = - 4$ — Check the term before: it is negative, so 25 really is the smallest positive one.

Common mistake

Answering 15. That is the position. The question asks for the number in the sequence, 25.

Second Differences (Extended/Higher)

If the first differences change but the second differences are equal, the sequence is quadratic. Its nth term has this form:

Formula

n th term = a n^{2} + bn + c, a = \frac{second difference}{2}

$a$ is the number in front of $n^{2}$ , $b$ the number in front of $n$ , and $c$ the number on its own. $b$ and $c$ can be 0.

Worked example

1
$3, 9, 19, 33$ — The sequence.
2
$6, 10, 14$ — First differences: take each term from the next. They are not equal, so it is not linear.
3
$4, 4$ — Second differences: the gaps between 6, 10, 14. They are equal, so it is quadratic.
4
$a = \frac{4}{2} = 2$ — Halve the second difference. The nth term starts 2n².
5
$2 n^{2} : 2, 8, 18, 32$ — Write out 2n² for n = 1, 2, 3, 4, so you can compare.
6
$3 - 2 = 1, 9 - 8 = 1, 19 - 18 = 1$ — Take 2n² from each term. The leftover is 1 every time.
7
$n th term = 2 n^{2} + 1$ — Add the leftover to $2 n^{2}$ .

Key idea

Write the differences down. Mark schemes give a mark for correct second differences, even if your final nth term is wrong.

Linear Leftover (Extended/Higher)

Often the leftover after taking away $a n^{2}$ is not one number but a linear sequence. Find its nth term too.

Worked example

1
$3, 6, 13, 24, 39$ — The sequence.
2
$3, 7, 11, 15$ — First differences. Not equal, so not linear.
3
$4, 4, 4$ — Second differences are equal, so it is quadratic.
4
$a = \frac{4}{2} = 2$ — Halve the second difference: the nth term starts $2 n^{2}$ .
5
$2 n^{2} : 2, 8, 18, 32, 50$ — Write out 2n² for n = 1 to 5.
6
$3 - 2 = 1, 6 - 8 = - 2, 13 - 18 = - 5$ — Take 2n² from each term. Watch the sign: 6 − 8 = −2.
7
$1, - 2, - 5, - 8, - 11$ — The leftovers. They go down by 3 each time, so they are linear.
8
$- 3 n + 4$ — Gap −3 gives −3n. At n = 1, −3n is −3, and 1 − (−3) = 4.
9
$n th term = 2 n^{2} - 3 n + 4$ — Put the two parts together.
10
$n = 3 : 2 (9) - 9 + 4 = 13$ — Check with n = 3. It gives 13, the 3rd term, so the answer is right.

Some textbooks use a shortcut instead. Match these to the start of the sequence:

Formula

a + b + c = 1st term, 3 a + b = 1st first difference, 2 a = second difference

Worked example

1
$2 a = 4 \Rightarrow a = 2$ — The second difference is 4. Divide by 2.
2
$3 (2) + b = 3 \Rightarrow b = - 3$ — The first first difference is 3. 6 + b = 3, so b = −3.
3
$2 + (- 3) + c = 3 \Rightarrow c = 4$ — The 1st term is 3. 2 − 3 = −1, and −1 + c = 3, so c = 4. Same answer.

Common mistake

Taking $2 n^{2}$ away from the differences instead of the terms. Always subtract from the sequence itself.

Cube Numbers

Cube numbers are 1, 8, 27, 64. Some sequences are the cubes moved up or down.

Worked example

1
$0, 7, 26, 63$ — The sequence. The gaps 7, 19, 37 grow fast, so try the cubes.
2
$1, 8, 27, 64$ — Write the cubes underneath: $1^{3}, 2^{3}, 3^{3}, 4^{3}$ .
3
$0 = 1 - 1, 7 = 8 - 1$ — Compare each term with the cube in the same place. Every term is 1 less.
4
$n th term = n^{3} - 1$ — So cube the position n, then take 1.
5
$10 th term = 1 0^{3} - 1 = 999$ — Put n = 10. $1 0^{3} = 1000$ , then take 1.

Key idea

If the terms are 0, 1, 8, 27, the cubes start one place late. The nth term is $(n - 1)^{3}$ .

Third Differences (Extended/Higher)

If the second differences are not equal either, go one level further. Equal third differences mean the sequence is cubic.

Formula

n th term = a n^{3} + \dots, a = \frac{third difference}{6}

$a$ is the number in front of $n^{3}$ . Why 6? The cubes 1, 8, 27, 64, 125 have third differences of 6.

Worked example

1
$- 1, 13, 51, 125, 247$ — The sequence.
2
$14, 38, 74, 122$ — First differences. Not equal.
3
$24, 36, 48$ — Second differences. Still not equal, so it is not quadratic.
4
$12, 12$ — Third differences are equal, so it is cubic.
5
$a = \frac{12}{6} = 2$ — Divide the third difference by 6. The nth term starts 2n³.
6
$2 n^{3} : 2, 16, 54, 128, 250$ — Write out 2n³ for n = 1 to 5.
7
$- 1 - 2 = - 3, 13 - 16 = - 3$ — Take 2n³ from each term. The leftover is −3 every time.
8
$n th term = 2 n^{3} - 3$ — Add the leftover to $2 n^{3}$ .

Key idea

Write the differences down. A correct row of third differences earns a mark on its own.

Which Method? Read the Question

Check the gaps first.

You see	You do
Equal gaps	see “The nth Term of a Linear Sequence”
Gaps grow by 2	compare with n²
Second differences equal	half of it times n²
Second differences not equal	find third differences
Third differences equal	divide by 6, times n³
Close to 1, 8, 27, 64	compare with n³
Is this number a term?	set equal and solve for n
Smallest positive term	solve > 0, round n up

Key idea

n must be a whole number. If it is not, the number is not a term.

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