lim n k = 1 n - 1 ( k - 1 ) ( n k - k 2 ) 2 r = 1 n r 3 - s = 0 n ( s 2 + ( n - s ) 2 ) = t then [ 43 t ] = _ _ _ _ ( [ . ] denotes GIF )

Question

$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n - 1} (k - 1) (n k - k^{2})}{2 \sum_{r = 1}^{n} r^{3} - \sum_{s = 0}^{n} (s^{2} + {(n - s)}^{2})} = t then$ $[43 t]$ $=____$ $([.] denotes GIF)$

Smart Achievers · Accepted Answer

(7)

$Required = \lim_{n \to \infty} \frac{3 {\sum_{r = 1}^{n - 1} (- r^{3} + r^{2} (n + 1) - n r)}}{n (n + 1) (3 n^{2} - n - 2)} = \frac{1}{6}$

Solution

Q.
$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n - 1} (k - 1) (n k - k^{2})}{2 \sum_{r = 1}^{n} r^{3} - \sum_{s = 0}^{n} (s^{2} + {(n - s)}^{2})} = t then$ $[43 t]$ $=____$ $([.] denotes GIF)$

Ans.
(7)

$Required = \lim_{n \to \infty} \frac{3 {\sum_{r = 1}^{n - 1} (- r^{3} + r^{2} (n + 1) - n r)}}{n (n + 1) (3 n^{2} - n - 2)} = \frac{1}{6}$

Want Us to Email you About Special Offers & Updates?

Solution

Q. limn→∞∑k=1n-1(k-1)(nk-k2)2∑r=1nr3-∑s=0n(s2+(n-s)2)=t then [43t]=____ ([.] denotes GIF)

Ans. (7) Required=limn→∞3{∑r=1n-1(-r3+r2(n+1)-nr)}n(n+1)(3n2-n-2)=16

Want Us to Email you About Special Offers & Updates?

Q.
$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n - 1} (k - 1) (n k - k^{2})}{2 \sum_{r = 1}^{n} r^{3} - \sum_{s = 0}^{n} (s^{2} + {(n - s)}^{2})} = t then$ $[43 t]$ $=____$ $([.] denotes GIF)$

Ans.
(7)

$Required = \lim_{n \to \infty} \frac{3 {\sum_{r = 1}^{n - 1} (- r^{3} + r^{2} (n + 1) - n r)}}{n (n + 1) (3 n^{2} - n - 2)} = \frac{1}{6}$