JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Sequences and Series

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If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eights terms is 243 , then the sum of the first 50 terms of this G.P. is:

(A) $\frac{1}{26} (3^{50} - 1)$ ✓ Correct
(B) $\frac{1}{13} (3^{50} - 1)$
(C) $\frac{2}{13} (3^{50} - 1)$
(D) $\frac{1}{26} (3^{49} - 1)$

Answer: (A)

Solution

Let the G.P. be

a, a r, a r^{2}, \dots

. The second, third and fourth terms sum to

a r (1 + r + r^{2}) = 3

, and the sixth, seventh and eighth to

a r^{5} (1 + r + r^{2}) = 243

. Dividing,

r^{4} = 81

, so

r = 3

for positive terms. Then

a \cdot 3 \cdot 13 = 3

gives

a = \frac{1}{13}

. The sum of the first fifty terms is

S_{50} = \frac{a ( r ^{50} - 1 )}{r - 1} = \frac{\frac{1}{13} ( 3 ^{50} - 1 )}{2} = \frac{1}{26} (3^{50} - 1)

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Single correct option

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JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Sequences and Series

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