If-the-sum-of-an-infinite-GP-be-3-and-the-sum-of-the-squares-of-its-term-is-also-3-then-its-first-term-and-common-ratio-are-

Question Number 43629 by peter frank last updated on 12/Sep/18

If the sum of an infinite GP be 3 and the

sum of the squares of its term is also 3,

then its first term and common ratio are

Answered by $@ty@m last updated on 13/Sep/18

A T Q,

a + a r + {a r}^{2} + \dots . . = 3

\Rightarrow \frac{a}{1 - r} = 3 \dots (1)

A g a i n,

{(a)}^{2} + {(a r)}^{2} + {({a r}^{2})}^{2} + {({a r}^{3})}^{2} + \dots \dots = 3

\Rightarrow a^{2} (1 + r^{2} + r^{4} + r^{6} + \dots .) = 3

\Rightarrow a^{2} \times \frac{1}{1 - r^{2}} = 3 \dots (2)

f r o m (1) & (2),

\frac{a}{1 - r} = \frac{a^{2}}{1 - r^{2}}

\Rightarrow a = 1 + r \dots (3)

f r o m (1) & (3)

\frac{1 + r}{1 - r} = 3

\Rightarrow r = \frac{1}{2} A n s .

a n d f r o m (3)

a = \frac{3}{2} A n s .

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