_ n 3 n 4+ (2+1 n )^2+ (2+2 n )^2+ + (3-1 n )^2 is equal to

MCQ

$\lim _{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to

A
$12$
B
$\frac{19}{3}$
C
$0$
✓
$19$

✓

Answer

Correct option: D.

$19$

d
$\lim _{n \rightarrow \infty} \frac{3}{n} \sum \limits_{r=0}^{n-1}\left(2+\frac{r}{n}\right)^2$

$=3 \int \limits_0^1(2+x)^2 d x=27-8=19$

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