+136 What is the largest integer n such that \((1 + 2 + 3 + \cdots+ n)^2 < 1^3 + 2^3 + \cdots+ 7^3?\)
The cube series on the right adds up to 784. The sum on the left, before squaring, adds up to [n(n + 1) / 2]
So we get [n(n +1) / 2]^2 < 784
=> n(n + 1) / 2 < 28 Square root both sides
=> n(n + 1) < 56 Multiply by 2 to break the fraction
=> n^2 + n - 56 < 0 Rewrite
=> (n -7)(n + 8) < 0 Factor
=> n < 7, n < -8 therefore the highest integer would be 6
In case you didn't know, a sum of cubes adds up to the [n(n + 1) / 2]^2 formula we derived from squaring the left-hand sum.
