Question

The 5th,9th and 16th terms of a linear sequence (A.P) are the consecutive terms of an exponential sequence (G.P)
(i) find the common difference of the linear sequence in terms of the first term
(ii)show that the 21st,37th and 65th terms of the linear sequence are also in an exponential sequence and hence find the common ratio

Answer 1

Pls help me solve

Answer 2

just use your formulas.
(a+8d)/(a+4d) = (a+15d)/(a+8d)
3a = 4d
d = 3/4 a
Now show that the other three terms also have a common ratio

Answer 3

for the AP,
term5 = a+4d
term9 = a+8d
term16 = a + 15d

but these are now in a GP
(a+8d)/(a+4d) = (a+15d)/(a+8d)
(a+8d)^2 = (a+4d)(a+15d)
a^2 + 16ad + 64d^2 = a^2 + 19ad + 60d^2
3ad - 4d^2 = 0
d(3a - 4d) = 0
d = 0 which does not give an interesting AP
or
d = 3a/4

repeat with:
(a+36d)^2 = (a+20d)(a+64d)
a^2 + 72ad +1296d^2 = a^2 + 84ad + 1280d^2
16d^2 - 12ad = 0
4d^2 - 3ad = 0
d = 3a/4 , same result

r = term9/term5 = (a+8d)/(a+4d)
= (a + 8(3a/4))/(a + 4(3a/4) = 7a/4a = 7/4