# Homework Help: Spivak's Calculus Chapter 2 Problem 1(ii)

1. Jan 12, 2014

### bassnbrats

1. The problem statement, all variables and given/known data
Prove the following formulas by induction.

(ii) 13+...+n3= (1+...+n)2.

I am starting Spivak's Calculus by myself, and I simply do not understand Spivak's proof for the sum of the cubes equaling the square of the sum, exercise 1 (ii) of chapter 2 in his third edition. I tried to attach his proof to this thread. The only step I cannot figure out is the first, where (1+...+k+[k+1])2 = (1+...+k)2+2(1+...+k)(k+1)+(k+1)2.

I have already worked through problem 1 (i), so I understand everything that follows the first line of the proof. But how/why does the square of the sums expand this way once induction is applied? Please help!

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Last edited: Jan 12, 2014
2. Jan 12, 2014

### SammyS

Staff Emeritus
What is (a + b)2 ?

Now let a = 1 + 2 +3 + ... + k

and b = (k + 1)

3. Jan 12, 2014

### bassnbrats

Oh bother...

thank you

4. Jan 12, 2014

### SammyS

Staff Emeritus
With all those + ... + k ... it can be hard to see.