I just need a hint to solve the problem. The method used is illustrated in the example problem as follows:

I am supposed to imitate the method used in the example problem to solve the challenge problem in the book as follows

Imitate the method used in the last problem to find a formula for the sum
1^{2} + 2^{2} + 3^{2} + ... + k^{2}

when k is a positive integer.

I have tried various things like (a+1)^{4} - a^{4}, (a+1)^{3} - a^{3}, triangle numbers, odd numbers, but I havent been able to solve it. I just need a hint. This is not a homework problem. This is self study.

and do summation on both side. By telescoping method, only 2 terms left in the left hand side. You need to do a little manipulation on the right hand side though.