Class 11

Math

Algebra

Binomial Theorem

The ratio of the coefficient of $(r+1)_{th}$ term in the expansion of $(1+x)_{n+1}$ to the sum of the coefficients of $r_{th}$and $r+1_{th}$ terms in the expansion of $(1+x)_{n}$ is:

- $1:1$
- $1:2$
- $2:1$
- $1:4$

$T_{r+1}=_{n+1}C_{r}(1)_{n+1−r}x_{r}=_{n+1}C_{r}x_{r}$

Thus the coefficient of $(r+1)_{th}$ term is $_{n+1}C_{r}$

Now,

$(r+1)$th term in the expansion of $(1+x)_{n}$ is

$T_{r+1}=_{n}C_{r}(1)_{n−r}x_{r}$

So, coefficient of $(r+1)_{th}$ term is $_{n}C_{r}$

and thus the coefficient of $r$th term is $_{n}C_{r−1}$

$⇒_{n}C_{r}+_{n}C_{r−1}=_{n+1}C_{r}$ using standard formula

Hence, the required ratio is $1:1$.

and thus the coefficient of $r$th term is $_{n}C_{r−1}$

$⇒_{n}C_{r}+_{n}C_{r−1}=_{n+1}C_{r}$ using standard formula

Hence, the required ratio is $1:1$.