Question

If the value of $C_0 + 2 \cdot C_1 + 3 \cdot C_2 + \dots+ (n+1) \cdot C_n=576$ ,then n is equal to

• A. 7
• B. 5
• C. 6
• D. 9

Answer 1

The Correct Option is A

Given, $C_{0}+2 C_{1}+3 C_{2}+\ldots+(n+1) C_{n}=576$
We know that,
$(1+x)^{n}={ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots+{ }^{n} C_{n} x^{n} $
$\Rightarrow x(1+x)^{n}={ }^{n} C_{0} x+{ }^{n} C_{1} x^{2}+{ }^{n} C_{2} x^{3}+\ldots +{ }^{n} C_{n} x^{n+1}$
On differentiating w.r.t. $x$, we get
$(1+x)^{n}+x \cdot n(1+x)^{n-1} $
$={ }^{n} C_{0}+2 \cdot{ }^{n} C_{1} \cdot x+3{ }^{n} C_{2} x^{2}+\ldots+(n+1){ }^{n} C_{n} x^{n}$
On putting $n=1$, we get
$2^{n}+n \cdot 2^{n-1}={ }^{n} C_{0}+2 \cdot{ }^{n} C_{1}+3 \cdot{ }^{n} C_{1}+\ldots +(n+1){ }^{n} C_{n}$
$\Rightarrow 2^{n-1}(n+2)=576 \,\,\,\,\,\,$ (given)
$\Rightarrow 2^{n-1}(n+2)=2^{6} \times 9=2^{(7-1)} \cdot(7+2)$
On comparing, we get $n=7$