Let be a sequence whose odd-numbered terms constitute an arithmetic progression with…

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

(d_n)

be a sequence whose odd-numbered terms constitute an arithmetic progression with first term

p

and common difference

d

, while the even-numbered terms constitute a geometric progression with first term

q

and common ratio

r

where

|r|<1

. Answer the following questions without substituting numerical values. (i) Derive an expression for the sum of the first

2n

terms of the sequence in terms of

p, d, q, r

and

n

. (ii) Explain, with reasons, how the sum behaves as

n\to\infty

in each of the following cases: (a)

d>0

;

|r|<1

. (b)

d<0

;

0<|r|<1

. (iii) State the condition on

d

and

r

under which the series obtained by extending the sum to infinity will have a finite limit.