Question

S=1×2+2×5+3×10+....10×101

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Answer 1

                         $S=1(1^2+1)+2(2^2+1)+3(3^2+1)+\cdots+10(10^2+1)\\ \\ S=(1^3+1)+(2^3+2)+(3^3+3)+\cdots+(10^3+10)\\ \\ S=(1^3+2^3+3^3+\cdots+10^3)+(1+2+3+\cdots+10)\\ \\ S=\left(\dfrac{10\times 11}{2}\right)^2+\dfrac{10\times 11}{2}\\ \\ S=55^2+55\\ \\ \boxed{S=3080}$