Home / IGCSE Maths 0580 / Algebra and graphs / Quadratic, cubic and exponential sequences
Cambridge IGCSE · Topic 2.7
IGCSE Maths 0580 / Topic 2 / IGCSE

When the gaps grow.

Not every sequence goes up in equal steps. Square numbers, cube numbers and doubling all leave fingerprints in the differences, and once you can read them you can write the nth term.

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The key idea

Constant FIRST differences: linear, \(dn + c\). Constant SECOND differences: quadratic, and the coefficient of \(n^2\) is half the second difference. A constant RATIO between terms: exponential, \(a \times r^{\,n-1}\).

terms1st2nd 411223756 7111519 444 second difference 4, so the sequence starts 2n²
Fig. 1: for 4, 11, 22, 37, 56 the first differences change but the second differences are all 4, so the nth term is 2n² + n + 1
Section 01

Square and cube numbers.

In the lab, choose a quadratic sequence and watch the lab stack the first and second differences under the terms. Switch to a cubic and see how many rows of differences you need before they become constant.

\(n\)12345
\(n^2\)1491625
\(n^3\)182764125
Worked example Close to the squares
Find the nth term of \(2, 5, 10, 17, \ldots\)
  1. Differences 3, 5, 7 are not constant, so it is not linear.
  2. Compare with \(n^2\): \(1, 4, 9, 16\). Each term is 1 more.
  3. Check \(n = 3\): \(9 + 1 = 10\). Correct.
Answer: \(n^2 + 1\)
Worked example Close to the cubes
Find the nth term of \(0, 7, 26, 63, \ldots\)
  1. Compare with \(n^3\): \(1, 8, 27, 64\). Each term is 1 less.
  2. Check \(n = 2\): \(8 - 1 = 7\). Correct.
Answer: \(n^3 - 1\)
Worked example A multiple of the squares
Find the nth term of \(3, 12, 27, 48, \ldots\)
  1. Divide each term by 3: \(1, 4, 9, 16\), the square numbers.
  2. So each term is \(3 \times n^2\).
Answer: \(3n^2\)
Examiner trap

\(n^2\) means \(n \times n\), not \(2n\). For \(n = 5\), \(n^2 = 25\) but \(2n = 10\). Likewise \(3n^2\) means \(3 \times n^2\), not \((3n)^2\).

Section 02

Second differences.

(Extended) For a general quadratic \(an^2 + bn + c\), the second differences are all \(2a\). So halve the second difference to get \(a\), subtract \(an^2\) from every term, and what is left is a linear sequence.

Worked example Full quadratic (Extended)
Find the nth term of \(4, 11, 22, 37, 56, \ldots\) (Fig. 1)
  1. First differences: 7, 11, 15, 19. Second differences: 4, 4, 4.
  2. \(a = 4 \div 2 = 2\), so the nth term starts \(2n^2\).
  3. \(2n^2\) is \(2, 8, 18, 32, 50\). Subtract from the terms: \(2, 3, 4, 5, 6\).
  4. That leftover is linear with nth term \(n + 1\).
  5. Check \(n = 3\): \(18 + 3 + 1 = 22\). Correct.
Answer: \(2n^2 + n + 1\)
Worked example Half a square (Extended)
Find the nth term of the triangle numbers \(1, 3, 6, 10, 15, \ldots\)
  1. First differences 2, 3, 4, 5; second differences 1.
  2. \(a = \dfrac{1}{2}\), so subtract \(\dfrac{1}{2}n^2\): \(0.5, 2, 4.5, 8, 12.5\).
  3. Leftover: \(0.5, 1, 1.5, 2, 2.5\), which is \(\dfrac{1}{2}n\).
Answer: \(\dfrac{1}{2}n^2 + \dfrac{1}{2}n = \dfrac{n(n + 1)}{2}\)
Your turn
(Extended) Find the nth term of \(3, 10, 21, 36, \ldots\)
Examiner trap

The coefficient of \(n^2\) is HALF the second difference. Using the second difference itself (\(4n^2\) for Fig. 1) leaves a leftover that is not linear, and the formula fails at \(n = 2\).

Stage 1 · Learn

Check your understanding

Four quick checks. Each correct answer earns XP and records this skill on your profile.

Quick check+10 XP

What is the next term of \(1, 4, 9, 16, 25, \ldots\)?

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Find the nth term of \(2, 5, 10, 17, \ldots\)

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(Extended) Find the nth term of \(3, 6, 12, 24, \ldots\)

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(Extended) What are the second differences of the sequence \(2n^2 + 3n\)?

Section 03

Multiply each time.

(Extended) In the exponential tab of the lab, set a first term and a multiplier and watch the bars shoot up (multiplier above 1) or shrink towards zero (between 0 and 1). The nth term is the first term times the multiplier \(n - 1\) times.

Worked example Doubling (Extended)
Find the nth term of \(3, 6, 12, 24, \ldots\)
  1. Ratio: \(6 \div 3 = 2\), \(12 \div 6 = 2\). Constant ratio, so exponential with \(r = 2\).
  2. First term \(a = 3\). The nth term is \(3 \times 2^{\,n-1}\).
  3. Check \(n = 4\): \(3 \times 2^{3} = 24\). Correct.
Answer: \(3 \times 2^{\,n-1}\)
Worked example Halving (Extended)
Find the nth term of \(64, 32, 16, 8, \ldots\)
  1. Ratio \(\dfrac{1}{2}\), first term 64, so \(64 \times \left(\dfrac{1}{2}\right)^{n-1}\).
  2. Since \(64 = 2^6\), this simplifies to \(2^{6} \div 2^{\,n-1} = 2^{\,7-n}\).
Answer: \(64 \times \left(\dfrac{1}{2}\right)^{n-1} = 2^{\,7-n}\)
Your turn
(Extended) Find the 8th term of \(2, 6, 18, 54, \ldots\)
Examiner trap

\(3 \times 2^{\,n-1}\) is NOT \(6^{\,n-1}\). The power applies only to the 2; work it out first, then multiply by 3.

Section 04

Mixing sequences.

(Extended) Some sequences are a familiar sequence with a twist: one more than a power of 2, or a fraction whose top and bottom are both linear. Spot each part separately.

Worked example Powers of 2, shifted (Extended)
Find the nth term of \(3, 7, 15, 31, 63, \ldots\)
  1. Add 1 to each term: \(4, 8, 16, 32, 64\). These are \(2^2, 2^3, 2^4, \ldots\), so \(2^{\,n+1}\).
  2. Take the 1 away again.
Answer: \(2^{\,n+1} - 1\)
Worked example Fractions (Extended)
Find the nth term of \(\dfrac{1}{4}, \dfrac{2}{7}, \dfrac{3}{10}, \dfrac{4}{13}, \ldots\)
  1. Numerators \(1, 2, 3, 4\): nth term \(n\).
  2. Denominators \(4, 7, 10, 13\): difference 3, so \(3n + 1\).
Answer: \(\dfrac{n}{3n + 1}\)
Examiner trap

Do not cancel or simplify the fractions in the sequence before finding the pattern: \(\dfrac{2}{4}\) written as \(\dfrac{1}{2}\) hides the pattern in the numerators and denominators.

Stage 2 · Exam

Exam-style questions

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Finish the four checks above to unlock the exam questions
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(Extended) Find the nth term of \(5, 12, 23, 38, 57, \ldots\)

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(Extended) Find the 10th term of the sequence \(2, 6, 18, 54, \ldots\)

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(Extended) Find the nth term of \(\dfrac{1}{3}, \dfrac{2}{5}, \dfrac{3}{7}, \dfrac{4}{9}, \ldots\)

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(Extended) Find the nth term of \(3, 7, 15, 31, 63, \ldots\)

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Growing sequences, decoded.

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Go deeper · practice
Six original Cambridge-style questions
Square and cube patterns, second differences, exponential sequences, a combined sequence and a dot-pattern problem.