For n-0- the value of n C 0-22- n C 1-32- n C 2-42 n C 3- upto n -1 terms is kn n -1 - 2 n- then k -

Let S= ^n C_0 2^2· ^nC_1+3^2· nbsp; ^nC_2 4^2 ^ n C_3+ ⋯ upto (n+1) terms S=∑_r=0^n nbsp; ( 1)^r·(r+1)^2 ^n C_r =∑_r=0^n ( 1)^r·(r^2+2r+1) ^n C_r =∑_r=0^n ...

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Standard XII

Mathematics

Sum of Binomial Coefficients with Alternate Signs

For n>0, the ...

Question

For $n > 0$ , the value of $^{n} C_{0} - 2^{2} \cdot^{n} C_{1} + 3^{2} \cdot^{n} C_{2} - 4^{2}^{n} C_{3} \dots$ upto $(n + 1)$ terms is $k n (n + 1) \cdot 2^{n},$ then $k =$

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Solution

Let $S =^{n} C_{0} - 2^{2} \cdot^{n} C_{1} + 3^{2} \cdot^{n} C_{2} - 4^{2}^{n} C_{3} + \dots$ upto $(n + 1)$ terms
$S = n \sum r = 0 (- 1)^{r} \cdot (r + 1)^{2}^{n} C_{r}$
$= n \sum r = 0 (- 1)^{r} \cdot (r^{2} + 2 r + 1)^{n} C_{r}$
$= n \sum r = 0 (- 1)^{r} \cdot (r (r - 1) + 3 r + 1)^{n} C_{r}$
$= n \sum r = 0 (- 1)^{r} \cdot r (r - 1) \cdot^{n} C_{r} + 3 n \sum r = 0 (- 1)^{r} \cdot r C_{r} + n \sum r = 0 (- 1)^{r} \cdot^{n} C_{r}$
$= n \sum r = 2 (- 1)^{r} \cdot r (r - 1) \cdot^{n} C_{r} + 3 n \sum r = 1 (- 1)^{r} \cdot r C_{r} + n \sum r = 0 (- 1)^{r} \cdot^{n} C_{r}$
$= n (n - 1) n \sum r = 2 (- 1)^{r}^{n - 2} C_{r - 2} + 3 n n \sum r = 1 (- 1)^{r}^{n - 1} C_{r - 1} + (1 - 1)^{n}$
$= n (n - 1) n \sum r = 2^{n - 2} C_{r - 2} \cdot (- 1)^{r - 2} - 3 n n \sum r = 1^{n - 1} C_{r - 1} \cdot (- 1)^{r - 1} + 0$
$= 0 [∵ n \sum r = 0 n_{c_{r}} a^{r} = (1 + a)^{n}]$

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For n>0, the value of nC0−22⋅ nC1+32⋅ nC2−42 nC3⋯ upto (n+1) terms is kn(n+1)⋅2n, then k=

For $n > 0$ , the value of $^{n} C_{0} - 2^{2} \cdot^{n} C_{1} + 3^{2} \cdot^{n} C_{2} - 4^{2}^{n} C_{3} \dots$ upto $(n + 1)$ terms is $k n (n + 1) \cdot 2^{n},$ then $k =$