Let an A.P. and a G.P. each consist of positive terms. Their -th terms are equal, and…

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Let an A.P. and a G.P. each consist of positive terms. Their

k

-th terms are equal, and additionally the

(2k)

-th term of the A.P. equals the

(k+2)

-th term of the G.P. (i) Show that the common ratio

r

(

0 < r < 1

) of the G.P. and the integer

k

are related by

r^{k+2} - 2r^{k} + 1 = 0

. (ii) If

r = \frac{1}{2}

, determine all positive integers

k

for which such sequences can exist, and justify your conclusion.