If ∑ n = 1 5 1 n n + 1 n + 2 n + 3 = k 3 ,   then k is equal to:

Question

If &#8721; n = 1 5 1 n n + 1 n + 2 n + 3 = k 3 , &#160; then k is equal to:

Aicharya · Accepted Answer

T n = 1 3 n + 3 - n n ( n + 1 ) ( n + 2 ) ( n + 3 ) = 1 3 1 n ( n + 1 ) ( n + 2 ) - 1 n + 1 n + 2 n + 3 &#8721; n = 1 5 T n = 1 3 1 1 &#183; 2 &#183; 3 - 1 2 &#183; 3 &#183; 4 + 1 2 &#183; 3 &#183; 4 - 1 3 &#183; 4 &#183; 5 &#8230; + 1 5 &#183; 6 &#183; 7 - 1 6 &#183; 7 &#183; 8 = 1 3 1 6 - 1 6 &#183; 56 &#160; = 1 3 56 - 1 336 = 55 3 336 = k 3 &#8658; k = 55 336

If &#8721; n = 1 5 1 n n + 1 n + 2 n + 3 = k 3 , &#160; then k is equal to:

Solution

If ∑ n = 1 5 1 n n + 1 n + 2 n + 3 = k 3 , then k is equal to: