Question

The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.

Options

• `425/216`

• `429/216`

• `288/125`

• `280/125`

Answer 1

The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to `underlinebb(288/125)`.

Explanation:

Let P = `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + ....`  ...(i)

⇒ `P/6 = 1/6 + 5/6^2 + 12/6^3 + 22/6^4 + ....` ...(ii)

  Equation (i) – Equation (ii), we get

`P - P/6 = 1 + 4/6 + 7/6^2 + 10/6^3 + 13/6^4 + ...`

⇒ `(5P)/6 = 1 + 4/6 + 7/6^2 + 10/6^3 + 13/6^4 + ....`  ...(iii)

⇒ `(5P)/6^2 = 1/6 + 4/6^2 + 7/6^3 + 10/6^4 + ....`  ...(iv)

Equation (iii) – Equation (iv), we get

`(5P)/6 - (5P)/6^2 = 1 + 3/6 + 3/6^2 + 3/6^3 + 3/6^4 + ...`

⇒ `(5/6)^2P = 1 + 3(1/6 + 1/6^2 + 1/6^3 + ...)`

⇒ `(5/6)^2P = 1 + 3((1/6)/(1 - 1/6))`

⇒ `(5/6)^2P = 1 + 3/5 = 8/5`

⇒ P = `8/5 xx 36/25 = 288/125`