First slide

Binomial theorem for positive integral Index

Question

Sum of two middle terms in the expansion of $(1 + x)^{2 n - 1}$ is

Moderate

A

$^{2 n - 1} C_{n - 1}$
B

$^{2 n - 1} C_{n}$
C

$^{2 n} C_{n}$
D

$^{2 n} C_{n + 1}$

Solution

Two middle terms in the expansion of $(1 + x)^{2 n - 1}$ are $(\frac{2 n - 1 + 1}{2})$ th and $(\frac{2 n - 1 + 1}{2} + 1)$ th terms, that is, nth and (n+1)th terms.
Coefficients of the middle terms in the expansion of
$(1 + x)^{2 n - 1}$ are $^{2 n - 1} C_{n - 1}$ and $^{2 n - 1} C_{n}$ and their $sum =^{2 n - 1} C_{n - 1} +^{2 n - 1} C_{n} =^{2 n} C_{n}$

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