Let f: N rightarrow R be such that f(1)=1 and | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for $n \geq 2$. Let $S_n = \sum_{k=1}^{n} k f(k)$. Then $S_n = n(n+1) f(n)$. For

Vedclass · Accepted Answer

$1/500$

Vedclass · Answer

(C) Given $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for $n \geq 2$. Let $S_n = \sum_{k=1}^{n} k f(k)$. Then $S_n = n(n+1) f(n)$. For $n \geq 2$, $S_n = S_{n-1} + n f(n) = n(n+1) f(n)$. Substituting $S_{n-1} = (n-1)n f(n-1)$, we get $(n-1)n f(n-1) + n f(n) = n(n+1) f(n)$. Dividing by $n$ $(n \geq 2)$, we get $(n-1) f(n-1) + f(n) = (n+1) f(n)$. $(n-1) f(n-1) = n f(n) \Rightarrow f(n) = \frac{n-1}{n} f(n-1)$. For $n=2$, $f(2) = \frac{1}{2} f(1) = \frac{1}{2}$. For $n=3$, $f(3) = \frac{2}{3} f(2) = \frac{2}{3} 	imes \frac{1}{2} = \frac{1}{3}$. In general, $f(n) = \frac{1}{n}$. Thus, $f(500) = \frac{1}{500}$.

Let $f: N \rightarrow R$ be such that $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for all $n \in N, n \geq 2,$ where $N$ is the set of natural numbers and $R$ is the set of real numbers. Then, the value of $f(500)$ is

Similar Questions

The number of functions $f : \{1, 2, 3, 4\} \rightarrow \{ a \in \mathbb{Z} : |a| \leq 8 \}$ satisfying $f(n) + \frac{1}{n} f(n+1) = 1$ for all $n \in \{1, 2, 3\}$ is

If $f: R \rightarrow R$ is a differentiable function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$ and if $f^{\prime}(4)=24$ and $f^{\prime}(0)=3$,then $f(4)=$

If $f(x)$ is a polynomial function satisfying $f(x) \cdot f(\frac{1}{x}) = f(x) + f(\frac{1}{x})$ and $f(4) = 65$,then the value of $f(6)$ is:

Let $f: R \rightarrow R$ be a twice differentiable function such that $(\sin x \cos y)(f(2x+2y) - f(2x-2y)) = (\cos x \sin y)(f(2x+2y) + f(2x-2y))$ for all $x, y \in R$. If $f'(0) = \frac{1}{2}$,then the value of $24f''\left(\frac{5\pi}{3}\right)$ is:

If $f(x+2y, x-2y) = xy$, then $f(x, y)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module