1.9: A geometric series and a partial-sum ratio

4PM1/2/November/2020 — Question 7 · 12 marks

A geometric series has first term (x−3)\displaystyle (x-3), second term (x+1)\displaystyle (x+1) and third term (4x−2)\displaystyle (4x-2).
(a) Find the two possible values of x\displaystyle x.
(5)
Given that x<1\displaystyle x<1,
(b) show that the series is convergent.
(2)
The sum to infinity of the series is S\displaystyle S.
(c) Find the value of S\displaystyle S.
(2)
The sum of the first n\displaystyle n terms of the series is Sn\displaystyle S_n. Given that
SSn=256255,\frac{S}{S_n}=\frac{256}{255},
(d) find the value of n\displaystyle n.
(3)