Question

The sum `sum_(k = 1)^20k 1/2^k` is equal to ______.

पर्याय

• `2 - 3/2^17`

• `1 - 11/2^20`

• `2 - 11/2^19`

• `2 - 21/2^20`

Answer 1

The sum `sum_(k = 1)^20k 1/2^k` is equal to `underlinebb(2 - 11/2^19)`.

Explanation:

Let, S = `sum_(k = 1)^20k. 1/2^k`

S = `1/2 + 2. 1/2^2 + 3. 1/2^3 + .... + 20. 1/2^20`  ...(i)

`1/2S = 1/2^2 + 2. 1/2^3 + .... + 19 1/2^20 + 20 1/2^21`  ...(ii)

On subtracting equations (ii) by (i),

`S/2 = (1/2 + 1/2^2 + 1/2^3 + .... + 1/2^20) - 20 1/2^21`

= `(1/2(1 - 1/2^20))/(1 - 1/2) - 20. 1/2^21`

= `1 - 1/2^20 - 10. 1/2^20`

`S/2 = 1 - 11. 1/2^20`

⇒ S = `2 - 11. 1/2^19`

= `2 - 11/2^19`