Question

The first third and ninth terms of a linear sequence are the first three terms of an exponential sequence. The 7th term of the linear sequence is 14. Find the common difference of a linear sequence the common ratio of the exponential sequence and the sum of the fifth to ninth terms of the exponential sequence

Answer 1

Srry idk

Answer 2

So we are told that
(a+2d)/a = (a+8d)/(a+2d) and a+6d = 14 or a = 14-6d
a^2 + 4ad + 4d^2 = a^2 + 8ad
leaves with 4d^2 = 4ad
d = a

We are also told thta a+6d = 14 or a = 14-6d
14-6d = d
d = 2 , then a = 2

checking:
the AS would be 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ...
does 2, 6, 18 form a GP ?? Yes, with a common ratio of 3
The GP would be : 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, ....
It all checks out.

For the AP, a = 2, d = 2
For the GP, a = 2, r = 3
the sum of the fifth to ninth terms of the exponential sequence
= ar^4 + ar^8
= a(r^4 + r^8)
= 2(81 + 6561) = 13284

or just adding the fifth and ninth term of my listed sequence
= 162 + 13122
= 13284

YEAAAHHH