The sum of the first \(n\) terms of a series is \(\displaystyle \frac14(13n-n^2)\), where \(n\) is a positive integer.
Find an expression for \(u_n\), the \(n\)th term of the series.[2]
Show that this series is an arithmetic series.[2]
A convergent geometric series has first term 3 and fourth term \(24x^{-3}\), where \(x\in\mathbb{R}\), \(x\ne0\).
Find the range of values that \(x\) can take.[3]
Find, in terms of \(x\), the sum of the geometric series after but not including the 10th term. Give your answer in the form \(\displaystyle \frac{p}{x^m(x-q)}\), where \(m\), \(p\) and \(q\) are positive integers.[3]
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Answer:(a)(i) \(u_n=\frac{7-n}{2}\) (a)(ii) common difference \(-\frac12\) (b)(i) \(x<-2\) or \(x>2\) (b)(ii) \(\frac{3072}{x^9(x-2)}\), \(m=9\), \(p=3072\), \(q=2\)