Let (1 + x)n = Co + C1x + C2x2 +........ + Cnxn and (C1 /C0)+ (2.C2 /C1) + (3.C3 /C2) + ....+ (nCn/Cn-1) =n (n+1 /K ) The value of K is

Question

Let (1 + x)n = Co + C1x + C2x2 +........ + Cnxn and (C1 /C0)+ (2.C2 /C1) + (3.C3 /C2) + ....+ (nCn/Cn-1) =n (n+1 /K ) The value of K is (A)  (1 /2) (B) 2 (C)  (1 /3) (D) 3

Tardigrade Admin · Accepted Answer

(C1/C0)+2 (C2/C1)+3 (C3/C2) + .... (nCn/Cn-1) = (n/1)+2 ((n(n-1)/2)/n) +3 ((n(n-1)(n-2)/3 !)/(n(n-1))2 !)+...+(n⋅1/n) = n + (n - 1) + (n - 2) + .... + 1 = (n(n+1)/2) ∴ K = 2.

Q. Let $(1 + x)^{n} = C_{o} + C_{1} x + C_{2} x^{2} + ........ + C_{n} x^{n}$ and $\frac{C _{1}}{C _{0}} + \frac{2. C _{2}}{C _{1}} + \frac{3. C _{3}}{C _{2}} + .... + \frac{n C _{n}}{C _{n - 1}}$ = $n \frac{n + 1}{K}$ The value of K is

$\frac{1}{2}$

2

$\frac{1}{3}$

3

Q. Let (1+x)n=Co​+C1​x+C2​x2+........+Cn​xn and C0​C1​​+C1​2.C2​​+C2​3.C3​​+....+Cn−1​nCn​​ =nKn+1​ The value of K is

21​

2

31​

3

C0​C1​​+2C1​C2​​+3C2​C3​​+....Cn−1​nCn​​ =1n​+2n2n(n−1)​​ +32!n(n−1)​3!n(n−1)(n−2)​​+...+nn⋅1​ =n+(n−1)+(n−2)+....+1 =2n(n+1)​ ∴K=2.

Q. Let $(1 + x)^{n} = C_{o} + C_{1} x + C_{2} x^{2} + ........ + C_{n} x^{n}$ and $\frac{C _{1}}{C _{0}} + \frac{2. C _{2}}{C _{1}} + \frac{3. C _{3}}{C _{2}} + .... + \frac{n C _{n}}{C _{n - 1}}$ = $n \frac{n + 1}{K}$ The value of K is

$\frac{1}{2}$

$\frac{1}{3}$