2025 NJC P1 Q6
2025 NJC P1 Q6
8 marks
- An infinite geometric series \(S\) has first term \(1\) and non-zero common ratio \(r\). It is given that the sum to infinity of \(S\) is equal to the square of the sum of the first three terms of \(S\).
- Show that \(r\) satisfies the equation \(r^4 + ar^3 + br^2 + cr + d = 0\), where \(a, b, c,\) and \(d\) are constants to be determined.[3]
- Find the possible values of \(r\).[1]
- An arithmetic progression with \(4n\) terms has first term \(7\) and common difference \(d\). Every \(4^{\mathrm{th}}\) term is removed. Find the sum of the remaining terms in terms of \(n\) and \(d\).[4]
Solution:
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Answer:(a)(i) \(a=b=1,c=d=-1\) (a)(ii) \(r=-0.661\) or \(r=0.848\) (3 s.f.) (b) \(6dn^2-3dn+21n\)
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