Further mathematics
posted by Tosin .
An exponential sequence of positive terms and a linear sequence have the same first term.the sum of their first terms is 3 the sum of their second terms is 3/2,and the sum of their third terms is 6.find their fifth terms.

Since the two sequences have the same first term, and they sum to 3, they both start with 3/2.
Let the arithmetic sequence be 3/2, 3/2 + d, 3/2 + 2d, ...
Let the geometric sequence be 3/2, 3/2 r, 3/2 r^2, ...
3/2 + d + 3/2 r = 3/2
so, d = 3/2 r
3/2 + 2d + 3/2 r^2 = 6
r = 3 or 1
d = 9/2 or 3/2
AS: 3/2, 3, 15/2, ...
GS: 3/2, 9/2, 27/2, ...
or
AS: 3/2, 3, 9/2, ...
GS: 3/2, 3/2, 3/2, ...
I assume you can make it to the 5th terms of each sequence.
Cool problem! 
105