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IGCSE · Practice questions · Quadratic, cubic and exponential sequences

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Six original Cambridge-style questions on non-linear sequences: square and cube numbers, quadratic nth terms, exponential sequences and combinations.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
The method

Differences and ratios tell you the type.

01
Recall
[2 marks]

(a) Write down the next two terms of \(1, 8, 27, 64, \ldots\) (b) Write down its nth term.

  • (a) \(5^3 = 125\), \(6^3 = 216\) ✓
  • (b) \(n^3\) ✓
02
Recall
[2 marks]

Find the nth term of \(0, 3, 8, 15, 24, \ldots\)

  • Compare with \(n^2\): \(1, 4, 9, 16, 25\). Each term is 1 less. ✓
  • \(n^2 - 1\) ✓
03
Calculation
[3 marks]

(Extended) Find the nth term of \(6, 15, 28, 45, 66, \ldots\)

  • First differences 9, 13, 17, 21; second differences 4, so \(a = 2\). ✓
  • Subtract \(2n^2\) (2, 8, 18, 32, 50): \(4, 7, 10, 13, 16\), which is \(3n + 1\). ✓
  • nth term \(2n^2 + 3n + 1\). ✓
04
Calculation
[3 marks]

(Extended) A sequence begins \(80, 40, 20, 10, \ldots\) (a) Find the nth term. (b) Find the first term that is less than 0.1.

  • (a) Ratio \(\dfrac{1}{2}\): \(T_n = 80 \times \left(\dfrac{1}{2}\right)^{n-1}\). ✓
  • (b) Keep halving: \(T_{10} = 80 \div 2^{9} = 0.156\,25\), \(T_{11} = 80 \div 2^{10} = 0.078\,125\). ✓
  • The first term below 0.1 is \(T_{11} = 0.078\,125\). ✓
05
Analysis
[4 marks]

(Extended) Sequence A has nth term \(n^2\) and sequence B has nth term \(2n + 3\). Each term of sequence C is the term of A minus the term of B.

(a) Write down the nth term of C. (b) Which term of C is equal to 60?

  • (a) \(n^2 - (2n + 3) = n^2 - 2n - 3\). ✓
  • (b) \(n^2 - 2n - 3 = 60\), so \(n^2 - 2n - 63 = 0\). ✓
  • \((n - 9)(n + 7) = 0\), so \(n = 9\) (\(n\) cannot be negative). ✓
  • Check: \(81 - 18 - 3 = 60\). It is the 9th term. ✓
06
Analysis
[5 marks]

Pattern \(n\) is a rectangle of dots with \(n\) rows and \(n + 1\) columns. So pattern 1 has 2 dots, pattern 2 has 6 and pattern 3 has 12.

(a) Write down an expression for the number of dots in pattern \(n\). (b) How many dots are in pattern 10? (c) Which pattern has 600 dots? (d) Show that no pattern has exactly 500 dots.

  • (a) \(n(n + 1) = n^2 + n\). ✓
  • (b) \(10 \times 11 = 110\). ✓
  • (c) Try values near \(\sqrt{600} \approx 24.5\): \(24 \times 25 = 600\). Pattern 24. ✓
  • (d) \(21 \times 22 = 462\) and \(22 \times 23 = 506\). 500 lies between two consecutive patterns, so no pattern has 500 dots. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.