The sum of the first $n$ terms, where $n\geq1$, of a geometric series is given by \[S_n=\fr

Question

The sum of the first $n$ terms, where $n\geq1$, of a geometric series is given by $$S_n=\frac54\left[1-\left(-\frac13\right)^n\right].$$ Write down the value of the sum to infinity of the geometric series and find the least value of $n$ such that the sum of the first $n$ terms and the sum to infinity differ by less than $10^{-3}$.[4] Given that $\displaystyle u_k=k\sum_{n=1}^{k}S_n$, find an expression for $u_k$ in terms of $k$ where $k\in\mathbb{Z}^+$.[4]

Accepted Answer

(a) $S_\infty=\frac54$, least $n=7$ (b) $u_k=\frac{5k}{16}\left\{4k+1-(-\frac13)^k\right\}$

2026 HCI P1 Q4

2026 HCI P1 Q4

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