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JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Sequences and Series

Let

a_{1}, a_{2}, a_{3}, \dots

be an A.P. If

\sum_{r = 1}^{\infty} \frac{a _{r}}{2 ^{r}} = 4

, then

4 a_{2}

is equal to ____ .

Answer: 16

Solution

Let the A.P. have first term

a_{1}

and common difference

d

. Then:

S = r = 1 \sum \infty \frac{a _{r}}{2 ^{r}} = r = 1 \sum \infty \frac{a _{1} + ( r - 1 ) d}{2 ^{r}} = 4

Split the sum:

S = a_{1} r = 1 \sum \infty \frac{1}{2 ^{r}} + d r = 1 \sum \infty \frac{r - 1}{2 ^{r}}

The first sum is

\frac{1/2}{1 - 1/2} = 1

. For the second sum, use

\sum_{r = 1}^{\infty} \frac{r}{2 ^{r}} = 2

, so

\sum_{r = 1}^{\infty} \frac{r - 1}{2 ^{r}} = 2 - 1 = 1

.

Thus:

S = a_{1} + d = 4

But

a_{2} = a_{1} + d

, so

a_{2} = 4

, and

4 a_{2} = 16

.

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Integer answer

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