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

identify the common ratio and apply the convergence condition |r|<1 for an infinite geometric series
calculate an infinite geometric sum and justify the result from the first term and common ratio
determine whether a repeated geometric process has a finite limit, solve a threshold for a minimum number of terms, and interpret the context

[Maximum number: 2]

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