# 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.

1. Steve

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

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