Let , , , and be real numbers with . For a positive integer ,…

MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

Let

a

b

c

, and

r

be real numbers with

r\neq1

. For a positive integer

n

, define (i)

Q_n=\displaystyle \sum_{k=1}^{n}(ak+b)+c\sum_{k=1}^{2n}r^{k-1}

. Derive a simplified closed-form expression for

Q_n

in terms of

a,\,b,\,c,\,r,

and

n

. (ii) Using the expression obtained in part (i), compute

Q_5

when

a=2,\,b=-1,\,c=3,

and

r=\tfrac{1}{2}

. Present your answer in rational form.