Mathematical induction problem.

[2sin54]

Hi. I am learning mathematical induction on my own and I came across this problem:

Homework Statement

Prove:
1*4 + 4*7 + 7*10 + ... + (3n - 2)(3n + 1) = n(n + 1)²2. The attempt at a solution

Quick test for n=1:

(3 -2)(3 + 1) = 1(1 + 1)²
4 = 4

Alright, so I rewrite this with, on the left side, after the '...' having two members:

1*4 + 4*7 + 7*10 + ... + (3n - 5)(3n - 2) + (3n - 2)(3n + 1) = n(n+1)²

Assume n = k and it stands for k:

1*4 + 4*7 + 7*10 + ... + (3k - 5)(3k - 2) + (3k - 2)(3k + 1) = k(k+1)²

Lets prove that it stands for n = k+1:

[1*4 + 4*7 + 7*10 + ... + (3k - 2)(3k + 1)] + (3k+1)(3k+4) = (k+1)(k+2)²

Now replace everything in brackets with k(k+1)² from the step above:

k(k+1)² + (3k+1)(3k+4) = (k+1)(k+2)²

k³ + 2k² + k + 9k² + 12k + 3k + 4 = (k+1)(k² + 4k + 4)

k³ + 11k² + 16k + 4 = k³ + 4k² + 4k + k² + 4k + 4 | -k³

11k² + 16k + 4 = 5k² + 8k + 4

6k² + 8k = 0Where's the mistake?

 

[SammyS]

Does it work for n = 2? No!

Added in Edit:

Try: 1*4 + 2*7 + 3*10 + 4*13 + ... (n)(3n+1) = n(n+1)² .

 

[ArcanaNoir]

It's not a true statement. That just goes to show you the proof by induction can be counted on :)

 

[2sin54]

Does it work for n = 2? No!

Added in Edit:

Try: 1*4 + 2*7 + 3*10 + 4*13 + ... (n)(3n+1) = n(n+1)² .

Now it is true. Oh, and should I keep testing values like up to n=3 or something? Because everywhere I look they just say that you have to test for n=1.

It's not a true statement. That just goes to show you the proof by induction can be counted on :)

So is my book lying? Because it asks to "Prove that these equalities are right/true"
Or is it the lack of my English skills and I didn't completely understand your post?

 

[ArcanaNoir]

By "not a true statement" I meant the left side of the equation did not equal the right side. You fixed it, now it does.

You don't have to keep testing for higher values than 1, unless the equality is only true for something starting higher up, like, it's true for n>6 or something, then you test it for the first true value and then do k+1.

 

[2sin54]

I still am missing something here... How could it be not a true statement if they write in the book that those equalities are true? I solved a supposed to be right equality and it showed up that it wasn't right. The exercise should then ask to Prove whether equalities are right or not.

 

[ArcanaNoir]

I just meant 1*4 + 4*7 + 7*10 + ... + (3n - 2)(3n + 1) = n(n + 1)² wasn't true for all n.
1*4 + 2*7 + 3*10 + 4*13 + ... (n)(3n+1) = n(n+1)^2 is true.