JEE Main · 2007 · 29 Apr 2007 · Offline · Sequences and Series

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In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals

(A) $\frac{1}{2} (1 - 5)$
(B) $\frac{1}{2} 5$
(C) $5$
(D) $\frac{1}{2} (5 - 1)$ ✓ Correct

Answer: (D)

Solution

Let the first term be

a

and the common ratio

r

. Each term equalling the sum of the next two means

a r^{n - 1} = a r^{n} + a r^{n + 1}

, and dividing by

a r^{n - 1}

gives

1 = r + r^{2}

. Solving

r^{2} + r - 1 = 0

gives

r = \frac{- 1 \pm 5}{2}

, and since all terms are positive the ratio must be the positive root

r = \frac{5 - 1}{2}

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option

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