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

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

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, $n t h$ 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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