# Proof by Induction

• Feb 16th 2007, 01:33 PM
Ideasman
Proof by Induction
Using Mathematical Induction:

Prove the sum of cubes of 3 consecutive pos. integers is divisble by 9.
• Feb 16th 2007, 01:46 PM
CaptainBlack
Quote:

Originally Posted by Ideasman
Using Mathematical Induction:

Prove the sum of cubes of 3 consecutive pos. integers is divisble by 9.

The sum of the cubes of 1, 2, 3 is 1+8+27=36 which is divisible by 9.

Now suppose that for some k: C(k)=k^3+(k+1)^3+(k+2)^3 is divisible by 9.

Then:

C(k+1)=C(k) - k^3 + (k+3)^3 = C(k) + 3 (k^2*3) + 3 (k*3^2) + 3^3

..............=C(k) + 9 [k^2 + 3k + 3]

hence as by assumption C(k) is divisible by 9 so is C(k+1), and as we have
established that C(1) is divisible by 9, we have established by mathematical
induction that C(n) is divisible by 9 for all positive integers n.

This proves (as it is the same thing in different words) that the sum of the
cubes of three consecutive positive integers is divisible by 9.

RonL
• Feb 16th 2007, 03:23 PM
Soroban
Hello, Ideasman!

This is Captain Black's proof . . . with my formatting.

Quote:

Using Mathematical Induction:
Prove the sum of cubes of 3 consecutive pos. integers is divisble by 9.

Verify S(1): .1³ + 2³ + 3³ .= .36 . . . divisible by 9.

Assume S(k): .k³ + (k+1)³ + (k+2)³ .= .9a . for some integer a.

Add (k+3)³ - k³ to both sides:

. . k³ + (k+1)³ + (k+2)³ + (k+3)³ - k³ .= . 9a + (k+3)³ - k³

. . (k+1)³ + (k+2)³ + (k+3)³ .= .9a + k³ - 9k² + 27k - 27 - k³

. . (k+1)³ + (k+2)³ + (k+3)³ .= .9a - 9k² + 27k - 27

. . (k+1)³ + (k+2)³ + (k+3)³ .= .9(a - k² + 3k - 3)

The left side is the left side of S(k+1); the right side is a multiple of 9.
. . The inductive proof is complete.

Corrected my error . . . thanks, Captain!
• Feb 17th 2007, 12:41 AM
CaptainBlack
Quote:

Originally Posted by Soroban
Hello, Ideasman!

This is Captain Black's proof . . . with my formatting.

Verify S(1): .1³ + 2³ + 3³ .= .36 . . . divisible by 9.

Assume S(k): .k³ + (k+1)³ + (k+2)³ .= .3a . for some integer a.

You need "k^3+(k+1)^3+(k+2)^3=9a" here, which is what you use later.

RonL
• Feb 20th 2007, 04:23 PM
Ideasman
Just a quick question-

I thought mathematicians don't like solving two solves of the equation at the same time to prove something... I thought you work with one side (IE: the (k + 1) side to show the other side; according to my math professor, it's bad to try solve a proof like this. I may be wrong..your thoughts?
• Feb 21st 2007, 08:43 AM
ThePerfectHacker
Quote:

Originally Posted by Ideasman
Just a quick question-

I thought mathematicians don't like solving two solves of the equation at the same time to prove something... I thought you work with one side (IE: the (k + 1) side to show the other side; according to my math professor, it's bad to try solve a proof like this. I may be wrong..your thoughts?

Corret, it is a logical error called taking the converse.