Question 1116659: A sequence is formed by adding corresponding terms of an A.P. and a
G. P., with r = 2, The first, second and third terms of the sequence are 3,7
and 12 respectively. Find the A.P. and the G.P., and give an expression for
the sum of the first n terms of the sequence.
Answer by greenestamps(13200)   (Show Source):
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In a problem like this, where the sequence formed by adding the AP and GP consists of whole numbers, it is almost certain that the numbers in both the AP and GP will be whole numbers.
For a GP with r=2, the most likely possibilities for the first three terms (so that the terms of the AP are positive) are 1, 2, 4 or 2, 4, 8. The first of these possibilities makes the terms of the other sequence 2, 5, and 8, which is an AP. The second makes the terms of the other sequence 1, 3, and 4, which is not an AP.
So the first three terms of the GP are 1, 2, 4; and the first three terms of the AP are 2, 5, 8.
The formula for the GP is  ; the formula for the AP is  .
So the formula for the combined sequence is
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