If the value of _ n _ k=0 ^ n-1 k n [ [m] k+1 n - [m] k n ],(m N) is equal to 1 10 then find the value of m .

Step-by-step solution

aligned L & = _ n _ k=0 ^ n-1 k n ( ( k+1 n )^ 1 m - ( k n )^ 1 m ]= _ n [ _ k=0 ^ n-1 k+1-1 n ( k+1 n )^ 1 m - _ k=0 ^ n-1 k n ( k n )^ 1 m ] & = _ n [ _ k=0 ^ n-1 k+1 n ( k+1 n )^ 1 m - _ k=0 ^ n-1 ( k n )^ 1+ 1 m - _ k=0 ^ n-1 1 n ( k+1 n )^ 1 m ] aligned If k+1=k Limit will become 1 n aligned & = _ n [ _ k=1 ^n k n ( k n )^ 1 m - _ k=0 ^ n-1 ( k n )^ 1+ 1 m - _ k=0 ^ n-1 1 n ( k+1 n )^ 1 m ] & = _ n (T_n-T₀ )- ₀^1 x^ 1 / m d x=1- m m+1 = 1 m+1 1 10 m=9 aligned