Grade 12 Maths · Number Patterns, Sequences and Series

Find the common ratio

Find the number each term is multiplied by in a geometric sequence by dividing a term by the one before, for example 3, 6, 12 gives r = 2.

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4)Gr 12 Geometric Sequence | IntroKevinmathscience

5) Solve X Geometric Sequence | IntroKevinmathscience

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Question 3.2 · 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

3.2 Calculate the sum of the areas of the first 10 circles.(4)

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More past paper questions: find the common ratio

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3. Question 3.1.1 · September 2024 · Paper 1 · Provincial · 3 marks
3.1 Given the geometric sequence: $8 (x - 2)^{2}; 4 (x - 2)^{3}; 2 (x - 2)^{4}; ...$ $x \neq = 2$
3.1.1 Determine the value(s) of x where the sequence converges.(3)
Show the memo (3 ticks)Memo
1. ✓
  Expression for r $r = \frac{x - 2}{2}$
  1 mark · Find the common ratio
2. ✓
  Convergence inequality $- 1 < \frac{x - 2}{2} < 1$
  1 mark · Use the convergence condition
3. ✓
  Answer $0 < x < 4$
  1 mark · Solve a linear inequality
4. Question 3.3.1 · March 2022 · Provincial · 3 marks
3.3 The following sequence forms a convergent geometric sequence: $\frac{3}{( x - 1 ) ^{2}} + \frac{1}{( x - 1 )} + \frac{1}{3} + \frac{( x - 1 )}{9} + \dots$
3.3.1 Determine the possible values of $x$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Determine common ratio $r = \frac{T _{2}}{T _{1}} = \frac{\frac{1}{x - 1}}{\frac{3}{( x - 1 ) ^{2}}} = \frac{1}{x - 1} \times \frac{( x - 1 ) ^{2}}{3} = \frac{x - 1}{3}$
  1 mark · Find the common ratio
2. ✓
  Apply convergence condition $- 1 < r < 1$
  $- 1 < \frac{x - 1}{3} < 1$
  1 mark · Use the convergence condition
3. ✓
  Solve for x $- 3 < x - 1 < 3$
  $- 2 < x < 4$
  1 mark · Solve a linear inequality
5. Question 3.3.1 · March 2021 · Provincial · 3 marks
3.3 Given: $\frac{24}{x} + 12 + 6 x + 3 x^{2} + \dots; x \neq = 0$
3.3.1 Determine the value of $x$ for which the series converges.(3)
Show the memo (2 ticks)Memo
1. ✓
  Find common ratio $r = \frac{12}{\frac{24}{x}} = \frac{12 x}{24} = \frac{x}{2}$
  1 mark · Find the common ratio
2. ✓
  Apply convergence condition $- 1 < r < 1$
  $- 1 < \frac{x}{2} < 1$
  $- 2 < x < 2$
  2 marks · Use the convergence condition
6. Question 3.2 · March 2021 · Provincial · 4 marks
3.2 Calculate: $\sum_{n = 3}^{\infty} 5 (3)^{1 - n}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Determine first term $n = 3 : T_{1} = 5 (3)^{1 - 3} = 5 (3)^{- 2} = \frac{5}{9}$
  1 mark · Read and write sigma notation
2. ✓
  Determine second term and ratio $n = 4 : T_{2} = 5 (3)^{1 - 4} = 5 (3)^{- 3} = \frac{5}{27}$
  $r = \frac{1}{3}$
  1 mark · Find the common ratio
3. ✓
  Apply sum to infinity formula $S_{\infty} = \frac{a}{1 - r} = \frac{\frac{5}{9}}{1 - \frac{1}{3}}$
  1 mark · Calculate the sum to infinity
4. ✓
  Final answer $= \frac{5}{6}$ (or $0, 83$ )
  1 mark · Calculate the sum to infinity

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