calculate-A-n-k-0-n-k-C-n-k-B-n-k-0-n-k-2-C-n-k-C-n-k-0-n-k-3-C-n-k-

Question Number 43007 by abdo.msup.com last updated on 06/Sep/18

c a l c u l a t e

A_{n} = \sum_{k = 0}^{n} k C_{n}^{k}

B_{n} = \sum_{k = 0}^{n} k^{2} C_{n}^{k}

C_{n} = \sum_{k = 0}^{n} k^{3} C_{n}^{k}

Commented by maxmathsup by imad last updated on 06/Sep/18

l e t p (x) = \sum_{k = 0}^{n} C_{n}^{k} x^{k} w e h a v e p (x) = {(x + 1)}^{n} a n d

p^{^{'}} (x) = \sum_{k = 1}^{n} C_{n}^{k} k x^{k - 1} \Rightarrow x p^{^{'}} (x) = \sum_{k = 0}^{n} C_{n}^{k} k x^{k} \Rightarrow \sum_{k = 0}^{n} {k C}_{n}^{k} = p^{'} (1) b u t

p^{^{'}} (x) = n {(x + 1)}^{n - 1} \Rightarrow p^{^{'}} (1) = n 2^{n - 1} = A_{n} a l s o w e h a v e

p^{^{'}} (x) + x p^{^{″}} (x) = \sum_{k = 1}^{n} k^{2} C_{n}^{k} x^{k - 1} \Rightarrow x p^{^{'}} (x) + x^{2} p^{^{″}} (x) = \sum_{k = 0}^{n} k^{2} C_{n}^{k} x^{k} \Rightarrow

\sum_{k = 0}^{n} k^{2} C_{n}^{k} = p^{^{'}} (1) + p^{^{″}} (1) b u t p^{^{″}} (x) = n (n - 1) {(x + 1)}^{n - 2} \Rightarrow

p^{″} (1) = n (n - 1) 2^{n - 2} \Rightarrow B_{n} = n 2^{n - 1} + n (n - 1) 2^{n - 2} a l s o w e h a v e

p^{^{'}} (x) + {x p}^{(2)} (x) + 2 x p^{(2)} (x) + x^{2} p^{(3)} (x) = \sum_{k = 1}^{n} k^{3} C_{n}^{k} x^{k - 1} \Rightarrow

{x p}^{^{'}} (x) + 3 x^{2} p^{(2)} (x) + x^{3} p^{(3)} (x) = \sum_{k = 0}^{n} k^{3} C_{n}^{k} x^{k} \Rightarrow

\sum_{k = 0}^{n} k^{3} C_{n}^{k} = p^{^{'}} (1) + 3 p^{(2)} (1) + p^{(3)} (1)

= n 2^{n - 1} + 3 n (n - 1) 2^{n - 2} + n (n - 1) (n - 2) 2^{n - 3} .

Answered by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

{(1 + x)}^{n} = {n C}_{0} x^{0} + {n C}_{1} x + {n C}_{2} x^{2} + {n C}_{3} x^{3} + \dots + {n C}_{n} x^{n}

d i f f e r d n t i a t e w . r . t x b o t h s i d e

n {(1 + x)}^{n - 1} = 0 + {n c}_{1} . 1 + {n c}_{2} . 2 x + {n c}_{3} . 3 x^{2} + . . + {n c}_{n} . {n x}^{n - 1}

p u t x = 1

n {(2)}^{n - 1} = 0 . {n c}_{0} + 1 . {n c}_{1} + 2 . {n c}_{2} + 3 . {n c}_{3} + . . + n . {n c}_{n}

n . 2^{n - 1} = \underset{k = 0}{\sum^{n}} k . {n c}_{k} \to a n s \to A

n {(1 + x)}^{n - 1} = 0 + {n c}_{1} . 1 + {n c}_{2} . 2 x + {n c}_{3} . 3 x^{2} + . . + {n c}_{n} . {n x}^{n - 1}

m u l t i p l y b o t h s i d e b y x

n x {(1 + x)}^{n - 1} = 0 . x + {n c}_{1} . 1 x + {n c}_{2} . 2 x^{2} + {n c}_{3} . 3 x^{3} + . . + {n c}_{n} . {n x}^{n}

d i f f e r e n t i a t e b o t h s i d e b y x

n (n - 1) x {(1 + x)}^{n - 2} + n {(1 + x)}^{n - 1} =

0 + {n c}_{1} . 1^{2} + {n c}_{2} . 2^{2} . x + {n c}_{3} . 3^{2} . x^{2} + . . + {n c}_{n} . n^{2} . x^{n - 1}

n o w p u t x = 1 b o t h s i d e

n (n - 1) . 2^{n - 2} + n . 2^{n - 1} = 0 + 1^{2} . {n c}_{1} + 2^{2} . {n c}_{2} + 3^{2} . {n c}_{3} + . . + n^{2} . {n c}_{n}

n (n - 1) . 2^{n - 2} + n . 2^{n - 1} = \underset{k = 0}{\sum^{n}} k^{2} . {n c}_{k} \to a n s B