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Chapter 7: Problem 532

\(\mathrm{C}_{0}+2 \mathrm{C}_{1}+3 \mathrm{C}_{2}+\cdots+(\mathrm{n}+1) \mathrm{C}_{\mathrm{n}}\) is \(=\) (a) \((\mathrm{n}+1) 2^{\mathrm{n}-1}\) (b) \((\mathrm{n}+2) 2^{\mathrm{n}-1}\) (c) \((\mathrm{n}+1) 2^{\mathrm{n}}\) (d) \((\mathrm{n}+1) 2^{\mathrm{n}-1}\)

Short Answer

Expert verified
The short answer is: \(\sum_{i=0}^n i \mathrm{C}_{n}= n(2)^{n-1}\), which corresponds to option (a) \((n+1)2^{n-1}\).

Step by step solution

01

Identity using binomial theorem

We will use the identity \((1+x)^n = \sum_{i=0}^n \mathrm{C}_{n}x^{i}\), which comes from the binomial theorem. We proceed by differentiating both sides of this identity with respect to x.
02

Differentiate with respect to x

Differentiating both sides of the identity, we get: \(\frac{d}{dx} (1+x)^n = \frac{d}{dx} \sum_{i=0}^n \mathrm{C}_{n}x^{i}\) This gives us: \(n(1+x)^{n-1} = \sum_{i=1}^n i \mathrm{C}_{n} x^{i-1}\)
03

Substitute x with 1

Now, we will substitute \(x=1\) into the differentiated equation to get: \(n(1+1)^{n-1} = \sum_{i=1}^n i \mathrm{C}_{n} 1^{1-1}\)
04

Simplify the equation

We can now simplify the equation: \(n(2)^{n-1} = \sum_{i=1}^n i \mathrm{C}_{n}\) Notice that the sum starts from \(i=1\) and we can add \(0*C_0\) to the left side to make it start from \(i=0\): \(n(2)^{n-1} = \sum_{i=0}^n i \mathrm{C}_{n}\) Comparing the result with the given options, we can conclude: \(\sum_{i=0}^n i \mathrm{C}_{n}= n(2)^{n-1}\), which corresponds to option (a) \((n+1)2^{n-1}\).

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Most popular questions from this chapter

If \(\mathrm{w} \neq 1\) is cube root of 1 then \({ }^{100} \sum_{\mathrm{r}=0}{ }^{100} \mathrm{c}_{\mathrm{r}}\left(2+\mathrm{w}^{2}\right)^{100-\mathrm{r}} \mathrm{w}^{\mathrm{r}}\) \(\begin{array}{llll}(\mathrm{a})-1 & \text { (b) } 0 & \text { (c) } 1 & \text { (d) } 2\end{array}\)

Co-efficient of \(\mathrm{x}^{5}\) in expansion of \((1+2 \mathrm{x})^{6}(1-\mathrm{x})^{7}\) is....... (a) 150 (b) 171 (c) 192 (d) 161

In the expansion of \(\left[\mathrm{a}^{(2 / 5)}+\mathrm{b}^{(1 / 3)}\right]^{35} \mathrm{a} \neq \mathrm{b}\), the number of terms in which the power of a and \(\mathrm{b}\) are integers are (a) 1 (b) 2 (c) 3 (d) 4

\(\left[5^{(1 / 2)}+7^{(1 / 8)}\right]^{1024}\) has number of rational terms \(=\) (a) 0 (b) 129 (c) 229 (d) 178

The constant term in expansion of \(\left[\left\\{(x+1) /\left\\{x^{(2 / 3)}-x^{(1 / 3)}+1\right\\}\right\\}-\left\\{(x-1) /\left(x-x^{(1 / 2)}\right)\right\\}\right]^{10}\) is (a) 210 (b) 105 (c) 70 (d) 35
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