Three numbers are in AP and their middle term is 6. If 2, 6 and 16 are added respectively to the first, second and third terms, the resulting numbers form a GP. If the common difference of the AP is positive, find the AP.
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Question

Three numbers are in AP and their middle term is 6. If 2, 6 and 16 are added respectively to the first, second and third terms, the resulting numbers form a GP. If the common difference of the AP is positive, find the AP.

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G1 Physics · Accepted Answer

Let the AP be 6−d,6,6+d. After addition the numbers are 8−d,12,22+d. For GP, 12²=(8−d)(22+d). Thus 144=176−14d−d² ⇒ d²+14d−32=0=(d−2)(d+16). Positive d=2. Hence the AP is 4,6,8.