JEE Main 2021 · Mathematics · Sequences and Series · hard

The sum of an infinite geometric progression is 15 and the sum of the squares of its terms is 45. Then its first term is

Numerical answer

The sum of an infinite geometric progression is

15

and the sum of the squares of its terms is

45

. Then its first term is

Show answer and solution

Answer: 5

Let the first term be

a

and ratio

r

. Then

\dfrac{a}{1-r}=15

and

\dfrac{a^2}{1-r^2}=45

. Substituting

a=15(1-r)

gives

\dfrac{225(1-r)}{1+r}=45

, so

5(1-r)=1+r

, giving

r=\tfrac23

. Hence

a=15\cdot\tfrac13=5

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