CAIE A-Level Mathematics 1.6.4 Geometric Convergence

Practise checking convergence and using exact first terms and common ratios to calculate sums to infinity or remaining tails.

Exam points

verify |r|<1 before applying the geometric sum-to-infinity formula
derive a and r from given terms, including exact surd values, before finding S∞
treat a removed-term tail as a new progression with its own first term and ratio

[Maximum number: 2]

A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by $50 \mathrm{~mm}, 40 \mathrm{~mm}$ and 32 mm respectively.

Find the greatest total depth in the ground which could theoretically be achieved.

$\frac{50}{1-\text { their } 0.8}$

M1

Use of sum to infinity formula with their r SOI.
Substituting a value of n into the sum formula M0.

=250

A1

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1.6.4—Geometric convergence question 1