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

1 : 1

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1 : 2

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2 : 1

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1 : 4

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Solution

The correct option is D $1 : 1$
$(r + 1)$ th term in the expansion is $(1 + x)^{n + 1}$ is
$T_{r + 1} =^{n + 1} C_{r} (1)^{n + 1 - r} x^{r} =^{n + 1} C_{r} x^{r}$
Thus the coefficient of $(r + 1)^{t h}$ 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 $(r + 1)^{t h}$ term is $^{n} C_{r}$
and thus the coefficient of $r$ th term is $^{n} C_{r - 1}$
$\Rightarrow^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ using standard formula
Hence, the required ratio is $1 : 1$ .

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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 rthand r+1th terms in the expansion of (1+x)n is:

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