<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">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 \(a &gt; 0\).</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(r\) and deduce that the geometric series is convergent.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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.</span><span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">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.</span><span class="inline-marks">[4]</span></div></li></ol>

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.

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.

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.

2025 ASRJC P1 Q8

2025 ASRJC P1 Q8