2024 IGCSE Additional Mathematics May/June 0606/12 Q10

12 marks

The first \(3\) terms of an arithmetic progression are \(3\sin2x,\ 5\sin2x,\ 7\sin2x\).
1. Show that the sum to \(n\) terms of this arithmetic progression can be written in the form \(n(n+a)\sin2x\), where \(a\) is a constant.[3]
2. Given that \(x=\dfrac{2\pi}3\), find the exact sum of the first \(20\) terms.[2]
The first \(3\) terms of a geometric progression are \(\ln2y,\ \ln4y^2,\ \ln16y^4\).
1. Find the \(n\)th term of this geometric progression.[2]
2. Find the sum to \(n\) terms of this geometric progression, giving your answer in its simplest form.[2]
The first \(3\) terms of a different geometric progression are \(\left(2w-\dfrac14\right),\ \left(2w-\dfrac14\right)^2,\ \left(2w-\dfrac14\right)^3\).
Find the values of \(w\) for which this geometric progression has a sum to infinity.[3]

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Answer:(a)(i) \(S_n=n(n+2)\sin2x,\ a=2\). (ii) \(-220\sqrt3\). (b)(i) \(u_n=2^{n-1}\ln2y\). (ii) \(S_n=(2^n-1)\ln2y\). (c) \(-\dfrac38<w<\dfrac58\).

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