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The ratio of the coefficient of (r+1)^{th} term in the expansi

Question

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

1 : 2

2 : 1

1 : 4

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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:

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Text solutionVerified

(r + 1)

th term in the expansion is

(1 + x)^{n + 1}

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}

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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Question Text	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:
Answer Type	Text solution:1
Upvotes	150