Question:
The total number of terms in the expansion of ${(x+a)^{47} - (x-a)^{47}}$ after simplification is
The total number of terms in the expansion of ${(x+a)^{47} - (x-a)^{47}}$ after simplification is
Updated On: Feb 14, 2025
- 24
- 47
- 48
- 96
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The Correct Option is A
Solution and Explanation
Total number of terms in $(x+a)^{n}-(x-a)^{n}$
$=\begin{cases}\frac{n+1}{2} & \text { if } n \text { is odd } \\ \frac{n}{2} & \text { if } n \text { is even }\end{cases}$
$\therefore$ Total number of terms in the expansion of
$(x+a)^{47}-(x-a)^{47}=\frac{47+1}{2} [\because n$ is odd $]$
$=\frac{48}{2} $
$=24 $
$=\begin{cases}\frac{n+1}{2} & \text { if } n \text { is odd } \\ \frac{n}{2} & \text { if } n \text { is even }\end{cases}$
$\therefore$ Total number of terms in the expansion of
$(x+a)^{47}-(x-a)^{47}=\frac{47+1}{2} [\because n$ is odd $]$
$=\frac{48}{2} $
$=24 $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.