Induction help

**monkbear** · May 26th 2011, 01:55 PM

Hi I need help in regards with proving this problem.

If the sequence is 1,13,37,73,121
then it should follow like this 1+12+24+36......12n+(12n+12)
For the formula 6n^2 - 6n + 1.

How would I go on to prove this with induction exactly? Thanks

**Plato** · May 26th 2011, 02:08 PM

Originally Posted by monkbear

Hi I need help in regards with proving this problem.
If the sequence is 1,13,37,73,121
then it should follow like this 1+12+24+36......12n+(12n+12)
For the formula 6n^2 - 6n + 1.
How would I go on to prove this with induction exactly? Thanks

Have a look at this Page

**bryangoodrich** · May 26th 2011, 03:08 PM

Originally Posted by monkbear

How would I go on to prove this with induction exactly? Thanks

Induction works in two steps. First you prove it true for some base case. In this case, if P(n) reads

$1 + 12 + 24 + ... + 12n + (12n + 12) = 6n^2 - 6n + 1$

show that P(0) is true. That should be straight-forward computation. The inductive step is to assume it is true for some $k \geq 0$ . Assuming P(k), demonstrate P(k+1). By establishing those two facts regarding P(n), you can prove it is true for every $n\geq 0$ . Do you understand why this two-step process works? In other words, do you know why the principle of mathematical induction (PMI) is a valid rule of inference?

Originally Posted by Plato

Have a look at this page.

That link didn't work for me Plato (using FF2). Broken?

**Plato** · May 26th 2011, 03:18 PM

Originally Posted by bryangoodrich

that link didn't work for me plato (using ff2). Broken?

fixed.

**bryangoodrich** · May 26th 2011, 04:14 PM

I've never used Wolfram Alpha before, but I have to say ... that's awesome!

**Archie Meade** · May 27th 2011, 05:20 AM

Originally Posted by monkbear

Hi I need help in regards with proving this problem.

If the sequence is 1,13,37,73,121
then it should follow like this 1+12+24+36......12n+(12n+12)
For the formula 6n^2 - 6n + 1.

How would I go on to prove this with induction exactly? Thanks

You should have a closer look at the pattern.

$u_1=1=1+12(0)$

$u_2=13=1+12(1)$

$u_3=37=1+12(1)+12(2)$

$u_n=1+12(1)+12(2)+....+12(n-1)$

To prove using induction that

$u_n=1+12(1)+12(2)+....+12(n-1)=6n^2-6n+1$

P(k)

$1+12(1)+12(2)+....+12(k-1)=6k^2-6k+1$

P(k+1)

$1+12(1)+12(2)+.....+12(k-1)+12k=6(k+1)^2-6(k+1)+1\;\;\;?$

Try to show that IF P(k) is true THEN P(k+1) will also be true.

Proof

If P(k) is true, then P(k+1) is

$1+12(1)+12(2)+...+12(k-1)+12k=6k^2-6k+1+12k=6k^2+6k+1$

which compares to

$6(k+1)^2-6(k+1)+1=6\left(k^2+2k+1\right)-6k-6+1$

$=6k^2+12k+6-6k-6+1$

$=6k^2+6k+1$

It remains to test the base case for n = 1, which is clearly true.

Math Help - Induction help

LinkBack

Thread Tools

Display

Induction help

Similar Math Help Forum Discussions

Strong induction vs. structural induction?

Induction

Rigorously prove the principle of complete induction from mathematical induction??

Mathemtical Induction Proof (Stuck on induction)

induction

Search Tags