Need mathematical Inudction Help (1 Viewer)

[blazer78]

i'm more concerned about 3 b) though. I can't seem to prove for n=k+1 (I hate factorials)

and this thing is due tomorrow. So need an answer in the next 3 hours.

Any help is appreciated. Thanks!

 

[darkliight]

a) What can't you do with this one? Give me a start and i'll help you finish.

b) After you prove it true for n = 1

Assume true for n = k: 1/2! + 2/3! + 3/4! + ... + k/(k+1)! = ((k+1)! - 1) / (k+1)!

Prove true for n = k + 1:

1/2! + 2/3! + 3/4! + ... + k/(k+1)! + (k+1) / (k+2)! = ((k+2)! - 1) / (k+2)!

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

(k+2)((k+1)! - 1) / (k+2)(k+1)! + (k+1) / (k+2)! = ((k+2)! - 1) / (k+2)! **multiplying the first fraction by (k+2) / (k+2)**

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

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

 

[Riviet]

Here's my solution to the main proof in 3 b) :
http://img114.imageshack.us/my.php?image=induction8us.png

 

[blazer78]

erm for 3 a) i can't prove true for step 3

@ darklight, as for 3 b) i don't see how you jumped from line 3 to line 4

 

[darkliight]

Headed to bed now, but here is a good start to your first problem.

After we know it works for n = 1, we assume it's true for k: 7^k - 3^k = 4A (A is an integer)

We need to show it's true for k + 1:

7^(k+1) - 3^(k+1)

7*7^(k) - 3*3^(k)

(4+3)*7^(k) - 3*3^(k)

4*7^(k) + 3*7^(k) - 3*3^(k)

It's all your to finish off, keeping in mind that 7^k - 3^k = 4A

 

[darkliight]

Line three:

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

The only part that changed, from this to line 4, was the first fraction: (k+2)((k+1)! - 1) / (k+2)(k+1)!
because I multiplied it by (k+2)/(k+2), I just simplified it.

So the numerator is now:
(k+2)((k+1)! - 1) = (k+2)(k+1)! - 1(k+2) = (k+2)! - k - 2

The denominator now is:
(k+2)(k+1)! = (k+2)!

The main thing here was knowing that, in general, (n+1)*n! = (n+1)!
so, for example, 3*2! = 3*2*1 = 3!
and in our case, (k+2)(k+1)! = (k+2)!

Hope that helps,
cheers

 

[blazer78]

thanks dudes, i'll be sure to keep that factorial rule in mind, it will come in handy.