You are Here: Home >< Maths

# FP1 QUESTION- proof. watch

1. Given that u1=0 and ur+1=2r−ur, use mathematical induction to prove that 2un=2n−1+(−1)^n,
nεℤ+.

I get the n=1 part.
I get the assumption part:

2k=nk-1+(-1)^k

how must you get K on its own?
the cd solution bank says:

2u(k+1)= 4k-2Uk= 4k-(2k-1+(-1)^k)

i can't seem to figure out how where they got 4 from?
2. , which is how the 4 arises. There's no need to rearrange for k.
3. All the assumption part involves is combining the two bits they gave you into a single expression.

You have U(k+1)= 2k-Uk and 2Uk = 2k-1+(-1)^k

Normally you'd just replace the Uk on the left with the formula, but since the formula is for 2Uk, you have to multiply the whole expression on the left by 2 so that you can substitute the formula in.

This gives -

2U(k+1)= 4k-2Uk

which is where their 4 comes from.

After that, you can put the formula in and rearrange the expression until you get the formula version of U(k+1).

2U(k+1)= 4k-(2k-1+(-1)^k)

2U(k+1)= 2k+1-1x(-1)^k

2U(k+1)= 2k+1+(-1)^(k+1)

or

2U(k+1)= 2(k+1)-1+(-1)^(k+1)

Which is the original formula they gave you, with k+1 instead of k, completing your proof.

Hope this helps.
4. (Original post by Alfalfas)
All the assumption part involves is combining the two bits they gave you into a single expression.

You have U(k+1)= 2k-Uk and 2Uk = 2k-1+(-1)^k

Normally you'd just replace the Uk on the left with the formula, but since the formula is for 2Uk, you have to multiply the whole expression on the left by 2 so that you can substitute the formula in.

This gives -

2U(k+1)= 4k-2Uk

which is where their 4 comes from.

After that, you can put the formula in and rearrange the expression until you get the formula version of U(k+1).

2U(k+1)= 4k-(2k-1+(-1)^k)

2U(k+1)= 2k+1-1x(-1)^k

2U(k+1)= 2k+1+(-1)^(k+1)

or

2U(k+1)= 2(k+1)-1+(-1)^(k+1)

Which is the original formula they gave you, with k+1 instead of k, completing your proof.

Hope this helps.

thanks a lot! =)
i really tend to get confused with proof.

### Related university courses

TSR Support Team

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

This forum is supported by:
Updated: January 31, 2010
Today on TSR

### He lied about his age

Thought he was 19... really he's 14

### University open days

Wed, 25 Jul '18
2. University of Buckingham
Wed, 25 Jul '18
3. Bournemouth University
Wed, 1 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams