First slide

Geometric progression

Question

A G.P consisting of terms with positive terms. If each term is equal to the sum of the next two terms, then the common ratio of G.P is

Moderate

A

$\frac{\sqrt{5} - 1}{2}$
B

$\frac{- \sqrt{5} - 1}{2}$
C

$\frac{- 1 \pm \sqrt{5}}{2}$
D

$\frac{\sqrt{5} + 1}{2}$

Solution

$Given that in a geometric progression, each term is equal to the sum of the next two terms it implies that a_{n} = a_{n + 1} + a_{n + 2} a r^{n - 1} = a r^{n} + a r^{n + 1} c a n c e l o u t t h e c o m m o n f a c t o r s w e g e t r^{2} + r - 1 = 0 S o l v i n g t h e q u a d r a t i c e q u a t i o n w e g e t r = \frac{- 1 \pm \sqrt{5}}{2} S i n c e a l l t h e t e r m s a r e p o s i t i v e t a k e p o s i t i v e c o m m o n r a t i o t h e r e f o r e, r = \frac{\sqrt{5} - 1}{2}$

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