First slide

Geometric progression

Question

Three numbers form a GP. If the 3rd term is decreased by 64, then the three numbers thus obtained will constitute an AP. If the second term
of this AP is decreased by 8, a GP will be formed again, then the numbers will be

Moderate

A

4,20,36
B

4,12,36
C

4,20,100
D

None of these

Solution

let a, ar, ar² are in GP and a, ar, ar² - 64 are in AP, we get
$a (r^{2} - 2 r + 1) = 64$ …….(i)
Again, a, ar -8, ar² - 64 are in GP.
$∴ (ar - 8)^{2} = a ({ar}^{2} - 64)$
$\Rightarrow a (16 r - 64) = 64$ …..(ii)
On solving Eqs.(i) and (ii), we get r =5,a =4
Thus, required numbers are 4, 20, 100.

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