Binomial Theorem Question - None

Question

Consider the following statements:
Statement-1: $\sum_{r=0}^{n}(r+1)\,{}^nC_r = (n+2)\,2^{n-1}$
Statement-2: $\sum_{r=0}^{n}(r+1)\,{}^nC_r\,x^r = (1+x)^n + nx(1+x)^{n-1}$
Which of the following is true?

Accepted Answer

Step 1: Verify Statement-2 algebraically

Using the identity $r\,{}^nC_r = n\,{}^{n-1}C_{r-1}$, we split:

$$\sum_{r=0}^{n}(r+1)\,{}^nC_r\,x^r = \sum_{r=0}^{n}r\,{}^nC_r\,x^r + \sum_{r=0}^{n}{}^nC_r\,x^r$$

Step 2: Evaluate the first sum

$$\sum_{r=0}^{n}r\,{}^nC_r\,x^r = \sum_{r=1}^{n}n\,{}^{n-1}C_{r-1}\,x^r = nx\sum_{r=1}^{n}{}^{n-1}C_{r-1}\,x^{r-1} = nx(1+x)^{n-1}$$

Step 3: Evaluate the second sum

$$\sum_{r=0}^{n}{}^nC_r\,x^r = (1+x)^n$$

Step 4: Combine results

$$\sum_{r=0}^{n}(r+1)\,{}^nC_r\,x^r = (1+x)^n + nx(1+x)^{n-1}$$

∴ Statement-2 is TRUE

Step 5: Verify Statement-1 using Statement-2

Setting $x=1$ in Statement-2:

$$\sum_{r=0}^{n}(r+1)\,{}^nC_r = 2^n + n\cdot2^{n-1} = 2^{n-1}(2+n) = (n+2)\cdot2^{n-1}$$

∴ Statement-1 is TRUE

Both statements are true.

Question:

Step-by-Step Solution

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