If `S_k=(1+2+3+...k)/k` then find the value of `S_1^2+S_2^2+....S_n^2`

1.	If `S_k=(1+2+3+...k)/k` then find the value of `S_1^2+S_2^2+....S_n^2`
Answer» `S_(k)=((1+2+3+...+k))/(k)=(k(k+1))/(2k)=(k+1)/(2).` `therefore sum_(k=1)^(n)S_(k)^(2)=sum_(k=1)^(n)(1)/(4)(k+1)^(2)=(1)/(4)(sum_(k=1)^(n)k^(2))+(1)/(2)(sum_(k=1)^(n)k)+(1)/(4)(1+1+ ... n " times")` `=(1)/(4)xx(1)/(6)n(n+1)(2n+1)+(1)/(4)n(n+1)+(1)/(4)n` `=(n)/(24)(2n^(2)+9n+13) " " `[on simplification].

Discussion