If _ n ( 1 n n+1 + 1 n n+2 + + 1 n n+n ) is equal to a b - c d , where a, b, c and d are positive integer, c and d are co-prime. Find the value of (a^4+b^3+c^2+d ) .

Step-by-step solution

aligned L & = _ n ( 1 n n+1 + 1 n n+2 + + 1 n n+n ) & = _ n _ r=1 ^n 1 n n+r = _ n _ r=1 ^n ( 1 n 1 n (1+ r n ) )= _ n _ r=1 ^n ( 1 n 1 1+ r n ) & = ₀^1 1 1+x d x=2 2 -2 aligned Hence a=2, b=2, c=2, d=1 a^4+b^3+c^2+d=29