The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is : Option 1)$24$Option 2)$32$Option 3)$36$Option 4)$28$

Selection of terms in G.P. -

If we have to take three terms in GP, we take them as $\frac{a}{r},a,ar$

- wherein

Extension : If we have to take (2K+1) term in GP, we take them as

$\frac{a}{r^{k}},\frac{a}{r^{k-1}},---------\frac{a}{r},a,ar------ar^{k}$

General term of an A.P. -

$T_{n}= a+\left ( n-1 \right )d$

- wherein

$a\rightarrow$ First term

$n\rightarrow$ number of term

$d\rightarrow$ common difference

$a, ar, ar^2$ be three terms.

$\\a\times ar \times ar^2 = 512\\ (ar)^3 = 512\\ ar =8$

$\frac{(a +4) + (ar^2)}{2} = ar + 4$

$a + ar^2 + 4 = 2ar + 8$

Putting $r = \frac{8}{a}$

$a + \frac{64}{a} = 20$

$\\ a^2 -20 + 64 = 0\\(a-16)(a-4) = 0\\ a =16, 4$

Numbers are 4,8,16 or 16, 8,4

Sum of numbers = 4+ 8 + 16 = 28

Option 1)

$24$

Option 2)

$32$

Option 3)

$36$

Option 4)

$28$

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