An infinite geometric progression has first term $a$ and common ratio $r$, where $a$ and

Question

An infinite geometric progression has first term $a$ and common ratio $r$, where $a$ and $r$ are non-zero. The sum of all the terms after the $n$th term of the progression is equal to twice the $n$th term. Show that the sum to infinity of the progression is three times the first term.[3] The positive integers, starting at $~1$, are grouped into sets, as follows. $\left\{ 1 \right\}$, $\left\{ 2,3 \right\}$, $\left\{ 4,5,6 \right\}$,… where there are $r$ integers in the $r$th set. Find, in terms of $r$, the first integer and the last integer in the $r$th set.[3] Prove that the sum of the integers in the $r$th set is $\frac{1}{2}r\left( 1+{{r}^{2}} \right)$.[2]

Accepted Answer

(a) $r=\frac23$, so $S_\infty=3a$ (b)(i) first integer $=\frac{r^2-r+2}{2}$, last integer $=\frac{r^2+r}{2}$

2023 YIJC P1 Q5

2023 YIJC P1 Q5

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