The total number of terms in the expansion of | vedclass

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Question

(A) The expansion of $(x+a)^n$ is given by $\sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k$. For $(x+a)^n - (x-a)^n$,the terms with even powers of $a$ cancel

Vedclass · Accepted Answer

$24$

Vedclass · Answer

(A) The expansion of $(x+a)^n$ is given by $\sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k$. For $(x+a)^n - (x-a)^n$,the terms with even powers of $a$ cancel out,leaving only the terms with odd powers of $a$. If $n$ is odd,the number of terms in the expansion of $(x+a)^n - (x-a)^n$ is given by $\frac{n+1}{2}$. Here,$n = 47$,which is an odd number. Therefore,the number of terms is $\frac{47+1}{2} = \frac{48}{2} = 24$.

The total number of terms in the expansion of $(x+a)^{47}-(x-a)^{47}$ after simplification is

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