Spivak's Calculus Chapter 2 Problem 1(ii)

[bassnbrats]

Homework Statement

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!

 

[SammyS]

Homework Statement

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!

What is (a + b)² ?

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

and b = (k + 1)

 

[bassnbrats]

Oh bother...

thank you

 

[SammyS]

Oh bother...

thank you

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