The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero,

Question

The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.[3]

Deduce that the geometric progression is convergent.[2]

Let $S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.[4]

Accepted Answer

The AP has common difference $d=-\frac{21}{2}$. The GP has common ratio $\frac12$, so it is convergent. The least value is $n=16$.

VJC P1 Q13

VJC P1 Q13

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