Question

The sum of the series 2.²⁰C₀ + 5.²⁰C₁ + 8.²⁰C₂ + 11.²⁰C₃ + .... + 62.²⁰C₂₀ is equal to ______.

Options

• 2²³

• 2²⁵

• 2²⁴

• 2²⁶

Answer 1

The sum of the series 2.²⁰C₀ + 5.²⁰C₁ + 8.²⁰C₂ + 11.²⁰C₃ + .... + 62.²⁰C₂₀ is equal to `underlinebb(2^25)`.

Explanation:

2.²⁰C₀ + 5.²⁰C₁ + 8.²⁰C₂ + 11.²⁰C₃ + .... + 62.²⁰C₂₀ 

`sum_(r = 0)^20(3r + 2) ""^20C_r` `((2","  5", " 8 ...A.P.),(2 + (n - 1)3))`

⇒  `sum_(r = 0)^20 3r ""^20C_r + 2sum_(r = 0)^20 20C_r`

⇒  3 × 20 × 2¹⁹ + 2 × 2²⁰ = 2¹⁹(60 + 4)

⇒  2¹⁹ × 2⁶ = 2²⁵

Where,
`sum_(r = 0)^20 ""^20C_r` = Sum of binomial coeff. = 2²⁰

 `sum_(r = 0)^20 ""^20C_r` = write (1 + x)²⁰ expansion, then differentiate and substitute x = 1