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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
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Solution
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` = tn+1
`"t"_(2("n" + 1))` = tn+1 and `"t"_("n" + + 1)` = tn+2
(i.e.) The middle terms are tn+1 and tn+2
tn+1 = `""^(2"n" + 1)"C"_"n"` and tn+2
= `"t"_(n" + 1 + 1)`
= 2n + 1Cn+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.
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