If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + c_{2} x^{n}$ Find the value of $3 C_{3} + 8 C_{4} + 15 C_{5} + 324 C_{6} + . . (n - 1) t e r m s$

Open in App

Solution

$T_{1} = (2^{2} - 1) C_{3}; T_{2} = (3^{2} - 1) C_{4}; T_{3} = (4^{2} - 1) C_{5} . . . .$
$∴ E = \sum_{r = 3}^{n} [(r - 1)^{2} - 1] C_{r} = \sum_{r = 3}^{n} [(r^{2} - 2 r)] C_{r} = \sum_{r = 3}^{n} r^{2} C_{r} - 2 \sum_{r = 3}^{n} r C_{r}$
We shall convert $\sum_{r = 3}^{n} t o \sum_{r = 0}^{n}$ by subtracting three terms corresponding to $r = 0, 1, 2,$ For $r = 0$ the term will be zero.
$∴ E = \sum_{r = 0}^{n} \cdot r^{2} C_{r} - (1^{2} \cdot C_{1} + 2^{2} C_{2}) - 2 \sum_{r = 0}^{n} r C_{r} - (1 \cdot C_{1} + 2 C_{2})$
$= n (n + 1) 2^{n - 2} - 1 - 4 C_{2} - 2 n 2^{n - 1} - 1 - 2 C_{2}$

Suggest Corrections

Similar questions

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}$ , then $\frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + . . . . . . . . + \frac{n C_{n}}{C_{n - 1}} =$

Q. If

(1 + x)^{n} = c_{0} + c_{1} x + c_{2} x^{2} + . . . . + c_{n} x^{n}

, find the value of

1^{2} c_{1} + 2^{2} c_{2} + 3^{2} c_{3} + . . . . . + n^{2} c_{n}

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}$ , then

$\frac{C_{1}}{C_{0}}$ + $\frac{2 C_{2}}{C_{1}}$ + $\frac{3 C_{3}}{C_{2}}$ + ........+ $\frac{n C_{n}}{C_{n - 1}}$ =

Q. Find the value of

C_{0} + \frac{C_{1} X}{2} + \frac{C_{2} X^{2}}{3} + . . . + \frac{C_{n} X^{n}}{n + 1}

Q. If

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . + C_{n} x^{n}

, then the value of

C_{0} + 2 C_{1} + 3 C_{2} + . . . . + (n + 1) C_{n}

will be

Join BYJU'S Learning Program

If (1+x)n=C0+C1x+C2x2+c2xn Find the value of 3C3+8C4+15C5+324C6+..(n−1)terms

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + c_{2} x^{n}$ Find the value of $3 C_{3} + 8 C_{4} + 15 C_{5} + 324 C_{6} + . . (n - 1) t e r m s$