Arithmetic and geometric progressions

Question code: 2934

Question 0404

A geometric series has common ratio

{r,}

and an arithmetic series has first term

{a}

and common difference

{d,}

where

{a}

and

{d}

are non-zero. The first three terms of the geometric series are equal to the first, tenth and thirteenth terms respectively of the arithmetic series.

Show that

{81 d^2 + 6 ad = 0.}

The sum of the first

{n}

terms of the arithmetic series is denoted by

{S.}

Given that

{a>0,}

find the set of possible values of

{n}

for which

{S}

exceeds

{4 a.}

Deduce that the geometric series is convergent, and find, in terms of

{a,}

the sum to infinity.

Answer

{\{ n \in \mathbb{Z}: 5 \leq n \leq 23 \}.}

The geometric series is convergent because

{-1 < r = \frac{1}{3} < 1.}

{S_\infty = \frac{3}{2} a.}

Question code: 2934