displaystyle limn→∞ ((12 +22 + ....+n2) √[n]n/ (n+1)(n+10)(n+100)) =

Question

displaystyle limn→∞ ((12 +22 + ....+n2) √[n]n/ (n+1)(n+10)(n+100)) =  (A) 3 (B) 1/3 (C) 2/3 (D)  ∞

Tardigrade Admin · Accepted Answer

displaystyle limn→∞ (n(n+1)(2n+1)√[n]n/6(n+1)(n+10)(n+100)) = displaystyle lim n→ ∞ ((2+(1/n)) displaystyle lim n→ ∞ √[n]n/6(1+(10)n)(1+(100)n) = (2/6) = (1/3)

Q. $n \to \infty lim \frac{( 1 ^{2} + 2 ^{2} + .... + n ^{2} ) n n}{( n + 1 ) ( n + 10 ) ( n + 100 )} =$

$3$

$1/3$

$2/3$

$\infty$

$n \to \infty lim \frac{n ( n + 1 ) ( 2 n + 1 ) n n}{6 ( n + 1 ) ( n + 10 ) ( n + 100 )}$
$= n \to \infty lim \frac{( 2 + \frac{1}{n} ) n \to \infty lim n n}{6 ( 1 + \frac{10}{n} ) ( 1 + \frac{100}{n} )} = \frac{2}{6} = \frac{1}{3}$

Q. n→∞lim​(n+1)(n+10)(n+100)(12+22+....+n2)nn​​=

3

1/3

2/3

∞

n→∞lim​6(n+1)(n+10)(n+100)n(n+1)(2n+1)nn​​ =n→∞lim​6(1+n10​)(1+n100​)(2+n1​)n→∞lim​nn​​=62​=31​

Q. $n \to \infty lim \frac{( 1 ^{2} + 2 ^{2} + .... + n ^{2} ) n n}{( n + 1 ) ( n + 10 ) ( n + 100 )} =$

$3$

$1/3$

$2/3$

$\infty$

$n \to \infty lim \frac{n ( n + 1 ) ( 2 n + 1 ) n n}{6 ( n + 1 ) ( n + 10 ) ( n + 100 )}$
$= n \to \infty lim \frac{( 2 + \frac{1}{n} ) n \to \infty lim n n}{6 ( 1 + \frac{10}{n} ) ( 1 + \frac{100}{n} )} = \frac{2}{6} = \frac{1}{3}$