the-third-sixth-and-seventh-terms-of-a-geometric-progression-whose-common-ratio-is-neither-0-nor-1-are-in-arithmetic-progression-prove-that-the-sum-of-the-first-three-terms-is-equal-to-fourth-

Question Number 103780 by bobhans last updated on 17/Jul/20

t h e t h i r d, s i x t h a n d s e v e n t h t e r m s o f a

g e o m e t r i c p r o g r e s s i o n (w h o s e c o m m o n

r a t i o i s n e i t h e r 0 n o r 1) a r e i n

a r i t h m e t i c p r o g r e s s i o n . p r o v e t h a t t h e

s u m o f t h e f i r s t t h r e e t e r m s i s e q u a l t o

f o u r t h .

Answered by bemath last updated on 17/Jul/20

A P : {a r}^{2}, {a r}^{5}, {a r}^{6} \Rightarrow 2 {a r}^{5} = {a r}^{2} + {a r}^{6}

2 r^{5} = r^{2} + r^{6} \Rightarrow 2 r^{3} = 1 + r^{4}

r^{4} - 2 r^{3} + 1 = 0; (r - 1) (r^{3} - r^{2} - r - 1) = 0

r^{3} = r^{2} + r + 1

n o w w e h a v e T_{4} = {a r}^{3}

s u m o f t h e f i r s t t h r e e t e r m s i s

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

= a (r^{3}) = T_{4} . p r o v e d

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