# Homework Help: A Challenge Problem in Techniques Of Problem Solving

1. Jun 18, 2010

### AK2

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
12 + 22 + 32 + ... + k2

when k is a positive integer.

I have tried various things like (a+1)4 - a4, (a+1)3 - a3, 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.

2. Jun 18, 2010

### ross_tang

Use

$$(k+1)^3 - k^3 = 3k^2+3k+1$$

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.

3. Jun 18, 2010

### taozhen

Why not like the Contiguous Numbers: for example,P=1+2+3+4+5....+100 and the answer of P is (1+100)*100/2 , so the S=(1+k)*k/2

4. Jun 18, 2010

### AK2

Thanks for the hint
I got my answer for S

S = (2k3+3k2+k)/6

5. Jun 18, 2010

### ross_tang

You are right. You may want to do some factorization also.

$$S = \frac{ n(n+1)(2n+1)}{6}$$