Question

the first, second, and fifth term of an AP are three consecutive terms of an exponential sequence. if the first term of the linear sequence is 7 ,find it common difference

Answer 1

How did they define "exponential sequence" in your course ?

Answer 2

Assuming you mean Geometric Sequence, we have

For The AP, a=7, so we want d.
The ratio between terms of a GP is constant, so we form the ratio between successive given terms of the AP:

(7+d)/7 = (7+4d)/(7+d)
d=14 or d=0

Now, d=0 is not very interesting:
AP = 7,7,7,7,7...
GP = ... 0,0,0, ...
so it doesn't work anyway.

With d=14, we have the AP

7,21,35,49,63
And the named terms form the terms of the GP
... 7, 21,63, ...
with r=3