Question

If the 17^th and 18^th terms in the expansion of (2 + a)⁵⁰ are equal, find the value of a.

Answer 1

Given : t₁₇ =  t₁₈

To Find : value of a

For (2 + a)⁵⁰ 

A = 2, b = a and n = 50

We have, t_r+1 = (_rⁿ) A^n-rb^r

For the 17^th term, r = 16 

t₁₇ = t₁₆₊₁

= (₁₆⁵⁰)(2)^50-16(a)¹⁶

= (₁₆⁵⁰)(2)³⁴(a)¹⁶

For the 18^th term, r = 17

t₁₈ = t₁₇₊₁

= (₁₇⁵⁰)(2)^50-17(a)¹⁷

= (₁₆⁵⁰)(2)³³(a)¹⁷

As 17^th and 18^th terms are equal

t₁₈ = t₁₇

• a = 1122 

Conclusion : value of a = 1122