Submit # The Wednesday Morning Math Challenge: Week 9 Answers

Nov 16 2016
John Urschel
Baltimore Ravens
Nov 16 2016

In Week 9, looked at three questions concerning the sum of cubes and the square of sums. (If you haven’t had a chance to look at them yet, click here.) The three questions build upon each other — and in the end, we get a little proof of one of number theory’s famous results.

1. We know that 1 + 2 + 3 + …. + k = k(k+1)/2. We can add k + 1 to both sides of the equation. Then we have

1 + 2 + 3 + …. + k + (k+1) = k(k+1)/2 + (k+1)

Now we can combine the terms on the righthand side.

k(k+1)/2 + (k+1) = k(k+1)/2 + 2(k+1)/2 = (k(k+1) +2(k+1))/2 = (k+1)(k+2)/2

Therefore, 1 + 2 + 3 + …. + k + (k+1) = (k+1)(k+2)/2

2. We’re given that 1^3 + 2^3 +….+k^3 = k^2(k+1)^2/4, and we want to show that 1^3 + 2^3 +…. + k^3 +(k+1)^3 = (k+1)^2(k+2)^2/4.

We can add (k+1)^3 to both sides of the original equation without violating the equality.

1^3 + 2^3 +….+k^3 + (k+1)^3 = k^2(k+1)^2/4 + (k+1)^3

As in the first problem, we can combine the terms on the righthand side, and then simplify the form.

k^2(k+1)^2/4 + (k+1)^3 = k^2(k+1)^2/4 + 4(k+1)^3/4 = (k^2(k+1)^2+4(k+1)^3)/4

= (k^2 + 4k + 4)(k+1)^2/4 = (k+2)^2(k+1)2/4

Therefore, 1^3 + 2^3 +…. + k^3 +(k+1)^3 = (k+1)^2(k+2)^2/4

3. Now we can use the results from problems 1 and 2 to solve problem 3.

We want to show that (1 +2 + 3 + … + n)^2 = 1^3 + 2^3 + 3^3 + …. n^3.

In the first problem, we were able to show that

1 + 2 + 3 + …. + k + (k+1) = (k+1)(k+2)/2

In the second, we showed that 1^3 + 2^3 +…. + k^3 +(k+1)^3 = (k+1)^2(k+2)^2/4

What do you notice about these two statements?

(k+1)^2(k+2)^2/4, from the second question, is equivalent to squaring (k+1)(k+2)/2, from the first.

That means that (k+1)^2(k+2)^2/4 is also equivalent to squaring the lefthand side of the first equation.

(1 + 2 + 3 + …. + k + (k+1))^2 = (k+1)^2(k+2)^2/4

Then we can substitute 1^3 + 2^3 +…. + k^3 +(k+1)^3 for (k+1)^2(k+2)^2/4, because we’ve already shown that they’re equivalent.

(1 + 2 + 3 + …. + k + (k+1))^2 = 1^3 + 2^3 +…. + k^3 +(k+1)^3. Our proof is complete.

This is what is known as a proof by induction. John Urschel
Baltimore Ravens