2024 IGCSE Additional Mathematics February/March 0606/12 Q9

12 marks

The first three terms of an arithmetic progression are \(\lg\theta^2\), \(\lg\theta^5\) and \(\lg\theta^8\).
1. Given that the sum to \(n\) terms of this progression is \(4732\lg\theta\), find the value of \(n\).[5]
2. This sum is equal to \(-14196\). Find the exact value of \(\theta\).[1]
The first three terms of a geometric progression are \(\lg\phi^3\), \(\lg\phi\) and \(\lg\phi^{\frac13}\).
1. Determine whether this geometric progression has a sum to infinity.[2]
2. Find the \(n\)th term of this geometric progression, giving your answer in the form \(3^A\lg\phi\), where \(A\) is a function of \(n\).[3]
3. Find the value of \(\phi\), given that the \(20\)th term is \(3^{-18}\).[1]

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Answer:(a)(i) \(n=56\) (ii) \(\theta=\dfrac1{1000}\) (b)(i) Yes, \(|r|=\dfrac13<1\). (ii) \(u_n=3^{2-n}\lg\phi\). (iii) \(\phi=10\).

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