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
Answers
Kit Kat
Srry idk
mathhelper
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
(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