Mathematics

Problem

(a) Derive an expression for the finite sum

T_n=\sum_{k=0}^{n-1} a\,r^{2k}

in terms of

a

r

and

n

, where

r\ne 1

. (b) Using your result from part (a), obtain the value of

T_n

when

a=7

r=\dfrac12

and

n=8

. (c) Show that when

|r|<1

, the sum

T_n

approaches

\dfrac{a}{1-r^2}

n\to\infty

, and verify this limiting value numerically for the data given in part (b) by comparing with

n=20