CAIE IGCSE Additional Math 12.5 Use Convergence and Sum to Infinity Questions

Practise deciding when a geometric progression converges, explaining why a sum to infinity exists and calculating that sum from its first term and ratio.

Exam points

Check the convergence condition |r| < 1 and use the sum-to-infinity formula for a convergent geometric progression.

[Maximum number: 5]

A geometric progression, A, has common ratio r, where |r|<1.
The terms of this progression are $a_1, a_2, a_3, \ldots$ .
Another geometric progression, B, has terms $b_1, b_2, b_3, \ldots$ , where
$b_1=a_2,\quad b_2=a_4,\quad b_3=a_6,\ldots$
The sum to infinity of A is $S_A$ and the sum to infinity of B is $S_B$ .
Find $\frac{S_B}{S_A}$ in terms of r.
Give your answer in its simplest form.

$S_A=\frac{a_1}{1-r}$ soi
Allow a for $a_1$ .
$r_b=r^2$ soi
$S_B=\frac{a_2}{1-r^2}=\frac{a_1r}{1-r^2}$
Allow a for $a_1$ with $r^2$ .
$\begin{aligned} k=\frac{S_B}{S_A}&=\frac{a_1r}{1-\text{their }r^2}\times\frac{1-r}{a_1}\\ \frac{S_B}{S_A}&=\frac{\frac{a_1r}{1-\text{their }r^2}}{\frac{a_1}{1-r}} \end{aligned}$
M1 dep on the first and second B1 for their $\frac{S_B}{S_A}$ ;
allow a for $a_1$ .
$\frac{r}{1+r}$ cao

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CAIE IGCSE Additional Math 12.5 Use Convergence and Sum to Infinity Questions question 1