The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric p

Question

The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with first term $a$ and common ratio $r$, where $r \neq 1$ and $a > 0$. Find the value of $r$ and deduce that the geometric series is convergent.[3] Using the value of $r$ found in (i), find the least value of $n$ such that the sum of all the terms after the $n^{\mathrm{th}}$ term of the geometric progression is less than $1$% of its sum of the first $n$ terms. [3] The sum, $S_n$, of the first $n$ terms of another sequence is given by $S_n = n \ln 2^{q(n+1)}$, where $q$ is a constant. Prove that the sequence follows an arithmetic progression.[4]

Accepted Answer

(a)(i) $r=\dfrac34$, so the series converges (ii) $n=17$ (b) The sequence is an AP with common difference $2q\ln2$

2025 ASRJC P1 Q8

2025 ASRJC P1 Q8

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