If [f:R R] is given by f(x) = x + 1, then the value of [ _n 1 n[ f(0) + f( 5 n ) + f( 10 n…

Question

If $$f:R \to R$$ is given by $f(x) = x + 1$, then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)} \right]$$ is :

Accepted Answer

$$f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)$$

$$ \Rightarrow 1 + 1 + {5 \over n} + 1 + {{10} \over n} + .... + 1 + {{5(n - 1)} \over n}$$

$$ \Rightarrow n + {5 \over n}{{(n - 1)n} \over 2} = {{2n + 5n - 5} \over 2} = {{7n - 5} \over 2}$$

$$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left( {{{7n - 5} \over 2}} \right) = {7 \over 2}$$

Solutions · 1