An arithmetic series has first term $a$ and common difference $d$, where $a0$ and \(d

Question

An arithmetic series has first term $a$ and common difference $d$, where $a>0$ and $d$ is non-zero. The first, eighth and thirteenth terms of the arithmetic series are equal to the first three terms of a geometric series respectively. Show that $a=-\frac{49}{2}d$.[2] Show that the geometric series is convergent and find its sum to infinity, $S$, in terms of $a$.[3] Given that the sum of the first $n$ terms of the arithmetic series is $25$ times the third term of the geometric series, find the value of $n$.[3] Find the least value of $k$ such that the sum of the first $k$ terms of the geometric series is more than $98\%$ of $S$.[3]

Accepted Answer

(a) $a=-\dfrac{49}{2}d$ (b) The geometric series is convergent, with $S=\dfrac{7a}{2}$ (c) $n=25$ (d) Least $k=12$

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