You are Here: Home

Proof by Induction

Maths and statistics discussion, revision, exam and homework help.

Announcements Posted on
1. Proof by Induction
Hi, could someone please explain, in stages how to complete a question I have been trying to finish for a while, thanks.
I'm new to this and cannot insert pictures, sorry.

Sum of (from 1 to n) r[3^(r-1)]=¼[3^n(2n-1)+1]

Thanks.
Last edited by helpstoask; 07-05-2012 at 17:18.
2. Re: Proof by Induction
I presume you can do the other bits so do you want to explain what you tried for your inductive step?
3. Re: Proof by Induction
Hi, could someone please explain, in stages how to complete a question I have been trying to finish for a while, thanks.
I'm new to this and cannot insert pictures, sorry.

Sum of (from 1 to n) r[3^(r-1)]=¼[3^n(2n-1)+1]

Thanks.
You need to show your working.

Inserting pictures isn't anything difficult, just upload the image on a image sharing site, and paste the link here.
4. Re: Proof by Induction
An outline of steps to get to the right answer will be:

Step 1: Show that it's true for ;

Step 2: Note that and apply the induction hypothesis to the big bracket;

Step 3: Rearrange stuff.

If you're still stuck, show your working and we'll see where you're going wrong.

[For typing maths into TSR, look at the Guide to LaTeX.]
Last edited by nuodai; 07-05-2012 at 17:30.
5. Re: Proof by Induction
Actually it isn't even true. Where did you get this from?
6. Re: Proof by Induction
(Original post by nuodai)
An outline of steps to get to the right answer will be:

Step 1: Show that it's true for ;

Step 2: Note that and apply the induction hypothesis to the big bracket;

Step 3: Rearrange stuff.

If you're still stuck, show your working and we'll see where you're going wrong.

[For typing maths into TSR, look at the Guide to LaTeX.]
The OP has written the question as: Sum of (from 1 to n) r[3^(r-1)]=¼[3^n(2n-1)+1]:

It means
7. Re: Proof by Induction
(Original post by raheem94)
The OP has written the question as: Sum of (from 1 to n) r[3^(r-1)]=¼[3^n(2n-1)+1]:

It means
He has now but he hadn't then!
8. Re: Proof by Induction
Hi guys, thanks for the replies, Raheem is right, the question is as he stated, sorry i could'nt put it in the normal form.
Thanks

And yes, it seems unbelievable, but I did originally type the question wrongly, sorry
9. Re: Proof by Induction
(Original post by raheem94)
The OP has written the question as: Sum of (from 1 to n) r[3^(r-1)]=¼[3^n(2n-1)+1]:

It means
The OP edited their post whilst I was writing my reply. I'll edit my post appropriately - thanks.
10. Re: Proof by Induction
I get to =¼[3^k(2k-1)+1+4(k+1)3^k]

Then I cannot continue
11. Re: Proof by Induction
There's actually a sneaky way to do this question.

Let , then and so .

So extending this to the sum we have

The sum in the brackets is a geometric series which evaluates to , which differentiates (by the quotient rule) to give . Plugging in and simplifying gives the desired result.

[But you've been asked to prove this by induction.]
Last edited by nuodai; 07-05-2012 at 17:41.
12. Re: Proof by Induction
I get to =¼[3^k(2k-1)+1+4(k+1)3^k]

Then I cannot continue
You're close, you just need to simplify. The +1 is fine because that's in the final answer; so take out as a common factor from the other bits and see if you can rearrange to get the required form.
13. Re: Proof by Induction
I get to =¼[3^k(2k-1)+1+4(k+1)3^k]

Then I cannot continue
That is correct.

Expand it all out then collect up like terms.

Factorise and remember that
14. Re: Proof by Induction
Still cannot, finish it, I don't know why, aiming for : ¼ [3^k+1 (2k+1) +1]
Sorry
15. Re: Proof by Induction
Still cannot, finish it, I don't know why, aiming for : ¼ [3^k+1 (2k+1) +1]
Sorry
Right, so you have

Take out the as a common factor to get

Can you see what to do now, given what you've been told so far?

Hint: simplify the bracket after and then look for common factors.
Last edited by nuodai; 07-05-2012 at 17:51.
16. Re: Proof by Induction
Still cannot, finish it, I don't know why, aiming for : ¼ [3^k+1 (2k+1) +1]
Sorry
17. Re: Proof by Induction
I get to =¼[3^k(2k-1)+1+4(k+1)3^k]

Then I cannot continue

Now try to simplify,
Last edited by raheem94; 07-05-2012 at 17:53.
18. Re: Proof by Induction
(Original post by nuodai)
You're close, you just need to simplify. The +1 is fine because that's in the final answer; so take out as a common factor from the other bits and see if you can rearrange to get the required form.
but how do you factorise out the 3^k when there is a +1 in that first term?
19. Re: Proof by Induction
(Original post by nuodai)
Right, so you have

Take out the as a common factor to get

Can you see what to do now, given what you've been told so far?

Hint: simplify the bracket after and then look for common factors.
I suggested expanding it all for a reason, students with less ability than you struggle with anything but the simplest factorisation.
Last edited by Mr M; 07-05-2012 at 17:53. Reason: Point proved!
20. Re: Proof by Induction
Ahh, thanks, I don't know why I kept getting stuck on that, i understand now, i kept getting to ¼[3^k(6k+4)]

Step 2: Register

Thanks for posting! You just need to create an account in order to submit the post
1. this can't be left blank

this is what you'll be called on TSR

2. this can't be left blank

never shared and never spammed

3. this can't be left blank

6 characters or longer with both numbers and letters is safer

4. this can't be left empty
1. By completing the slider below you agree to The Student Room's terms & conditions and site rules

2. Slide the button to the right to create your account

You don't slide that way? No problem.

Last updated: May 7, 2012
Study resources
Moderators

We have a brilliant team of more than 60 volunteers looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is moderated by:
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.
Post rating score:
These scores show if a post has been positively or negatively rated by our members.