The first, third and night terms of a linear sequence are the first three terms of an exponential sequence G. P if the seventh term of the linear sequence is 14.find the twentieth term of the linear sequence.

Answer 1

for the AS"
first term --- a
third term -- a+2d
ninth term -- a+8d

but those terms form a GP
(a+2d)/a = (a+8d)/(a+2d)
(a+2d)^2 = a(a+8d)
a^2 + 4ad + 4d^2 = a^2 + 8ad
4d^2 - 4ad = 0
d^2 - ad = 9
d(d-a) = 0
d = 0 ---> not too interesting
or
a = d

but term 7 of the AS = 14
a+6d = 14
a + 6a = 14
a = 2, then d = 2

term(20) = a + 19d = 2 + 38 = 40