IB Maths AI HL Ahl 1 11 Hl Infinite Geometric Sequences Questions

Practise finding infinite geometric sums when |r| < 1, deriving effective ratios and interpreting convergence in repeated-stage or applied models.

Exam points

Find sums of infinite geometric sequences and verify the convergence condition |r|<1.
Derive the effective common ratio when repeated number and size changes combine, then apply the infinite-sum formula.
Interpret convergence or divergence in a fractal, Markov-chain or repeated-stage model.

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Study AHL 1.11 (HL)—Infinite geometric sequences conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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Nanthana is completing an exploration on multiplication using complex numbers. She considers a geometric sequence where $u_{1}=9$ and $r=\frac{2}{3}+\frac{2}{3} \mathrm{i}$ .

Hence, find the sum of the infinite sequence $\left|u_{1}\right|,\left|u_{2}\right|,\left|u_{3}\right|, \ldots$ .

substitution of their r and 9 into the sum formula for infinite sequence
$S_{\infty}=\frac{9}{1-\frac{2 \sqrt{2}}{3}}$ $\left(S_{\infty}=\right) 157(157.367 \ldots, 81+54 \sqrt{2})$

Note: No FT if their $|r| \geq 1$ .

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IB Maths AI HL Ahl 1 11 Hl Infinite Geometric Sequences Questions question 1