I'm an eight-grader and I need help to answer this math problem (homework).
Problem:
Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$
Attempt:
I know how to calculate the quadratic sum using formula from here: Combination of quadratic and arithmetic series but how to calculate the cubic sum? How to calculate the series without using calculator? Is there any intuitive way like previous answer? Please help me. Grazie!
We have identities for sums of powers like these. In particular:
$$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$
$$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$
The rest is just a bit of arithmetic.