Question

SEQUENCES AND SERIES

QUESTION 1: The 8th and 20th terms of an arithmetic sequence are respectively equal to
the 6th and 8th terms of a geometric sequence. In the arithmetic sequence the first term is
a and the non-zero common difference d, whilst r is the common ratio in the geometric
sequence.
(a) Show that r^2=a+19d/a+7d
(b) If the 5th term of the arithmetic sequence is 4 and r is an integer, determine all
possible General Terms of both sequences.

Answer 1

Explanation:

As we Know that formula for arithmetic sequence is

`a_n=a_1+(d-1)`

and for geometric sequence is

`a_n=a1*r^(n-1)`

So,

According to given conditions

`a+7d=a*r^5 (i)\\ a+19d=a*r^7 (ii)`

By dividing equation (i) and (ii)

Hence proved that

`r^2=a+19d/a+7d`