Question

If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of (x + y)ⁿ are equal

Answer 1

Given n is odd.

So let n = 2n + 1

Where n is an integer.

The expansion (x + y)ⁿ has n + 1 terms.

= 2n + 1 + 1

= 2(n + 1) terms which is an even number.

So the middle term are `("t"^2("n"+ 1))/2` = t_n+1

`"t"_(2("n" + 1))` = t_n+1 and `"t"_("n" +  + 1)` = t_n+2

(i.e.) The middle terms are t_n+1 and t_n+2

t_n+1 = `""^(2"n" + 1)"C"_"n"` and t_n+2

= `"t"_(n" + 1 + 1)`

= ²n + ¹C_n+1

Now n + n + 1 = 2n + 1

⇒ `""^(2"n" + 1)"C"_"n" = ""^(2"n" + 1)"C"_("n" + 1)`

⇒ The coefficient of the middle terms in (x + y)ⁿ are equal.