Grade 12 Maths · Number Patterns, Sequences and Series

Find a geometric sequence term

Write or use Tn = a·rⁿ⁻¹ for a pattern that multiplies by the same number, for example 2, 6, 18, … gives T5 = 2 × 3⁴ = 162.

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Question 3.3 · November 2024 · Paper 1 · National (DBE) · 4 marks

A circle with radius

6

cm is drawn. A second, smaller circle is drawn through the centre of the first circle and also touches the first circle internally, as shown in the diagram. A third, smaller circle is drawn through the centre of the second circle and touches the second circle internally. The process of drawing circles continues and forms a geometric pattern. Diagram for this question

3.3 Which circle has a diameter of

\frac{3}{128}

cm?(4)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: find a geometric sequence term

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3. Question 2.2.2 · May/June 2025 · Paper 1 · National (DBE) · 2 marks
Given the arithmetic series: $5 + 7 + 9 + ... + 93$
2.2 The general term of a geometric sequence is $T_{n} = 2^{n + 2}$
2.2.2 Calculate $T_{20}$ (Write your answer as a power of $4$ .)(2)
Show the memo (2 ticks)Memo
1. ✓
  Calculate the 20th term $T_{20} = (2)^{20 + 2}$
  $T_{20} = 2^{22}$
  1 mark · Find a geometric sequence term
2. ✓
  Write as a power of 4 $2^{22} = (2^{2})^{11} = 4^{11}$
  1 mark · Change the base of a power
4. Question 1.2.2 · March 2024 · Paper 1 · Provincial · 2 marks
1.2 A frog is making a series of jumps. With every next jump, he has only enough energy left to jump $\frac{2}{3}$ the distance of his previous jump.
1.2.2 Determine the length of his ninth jump.(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute into geometric sequence formula $a = 81$
  $r = \frac{2}{3}$
  $n = 9$
  $T_{9} = 81 (\frac{2}{3})^{9 - 1}$
  1 mark · Find a geometric sequence term
2. ✓
  Calculate final length $T_{9} = \frac{256}{81} \approx 3, 16$ cm
  1 mark · Find a geometric sequence term
5. Question 1.1.2 · March 2021 · Provincial · 2 marks
1.1 The following geometric sequence is given: $10; 5; 2, 5; 1, 25; ...$
1.1.2 Determine the $n^{t h}$ term, $T_{n}$ , in terms of $n$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Determine general term formula $a = 10$ , $r = \frac{1}{2}$
  $T_{n} = 10 (\frac{1}{2})^{n - 1}$ or $T_{n} = 20 (\frac{1}{2})^{n}$
  2 marks · Find a geometric sequence term
6. Question 3.1 · March 2021 · Provincial · 5 marks
3.1 The second term of a geometric sequence is $\frac{5}{128}$ and the ninth term is $5$ . Determine the value of the common ratio.(5)
Show the memo (2 ticks)Memo
1. ✓
  Set up equations $a r = \frac{5}{128}$ ... (1)
  $a r^{8} = 5$ ... (2)
  2 marks · Find a geometric sequence term
2. ✓
  Solve for r $\frac{a r ^{8}}{a r} = \frac{5}{\frac{5}{128}}$
  $r^{7} = 128$
  $r^{7} = 2^{7}$
  $r = 2$
  3 marks · Solve an equation with rational exponents

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