The expression nCo+2 nC1 + 3 nC2+.....+(n+1)nCn is equal to

Question

The expression nCo+2 nC1 + 3 nC2+.....+(n+1)nCn is equal to (A)  (n+1)2n  (B)  2n(n+2)  (C)  (n+2)2n-1  (D)  (n+2)2n+1

Tardigrade Admin · Accepted Answer

We know, ( 1 + x)n = nC0 + nC1 x + nC2 x2 + ... + nCn xn ⇒ x(1+x)n = nC0 x + nC1 x2 + n C2 x3 + ... + nCn xn+1 On differentiating w.r.t. x, we get ( 1 + x )n + nx (1 + x)n-1 = nC0 + 2⋅ nC1 x + 3 ⋅ nC2 x2 + ... + (n + 1) ⋅ nCn xn Put x = 1, we get 2n + 2n-1⋅ n = nC1 + 2 ⋅ nC1 + 3 ⋅ nC2 + ... + (n + 1)⋅ nCn = ( n + 2)2n-1

Q. The expression $^{n} C_{o} + 2^{n} C_{1} + 3^{n} C_{2} + ..... + (n + 1)^{n} C_{n}$ is equal to

$(n + 1) 2^{n}$

$2^{n} (n + 2)$

$(n + 2) 2^{n - 1}$

$(n + 2) 2^{n + 1}$

Q. The expression nCo​+2nC1​+3nC2​+.....+(n+1)nCn​ is equal to

(n+1)2n

2n(n+2)

(n+2)2n−1

(n+2)2n+1

Q. The expression $^{n} C_{o} + 2^{n} C_{1} + 3^{n} C_{2} + ..... + (n + 1)^{n} C_{n}$ is equal to

$(n + 1) 2^{n}$

$2^{n} (n + 2)$

$(n + 2) 2^{n - 1}$

$(n + 2) 2^{n + 1}$