Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    0
    ReputationRep:
    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

    I do not get the induction part, some one please help?

    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?
    Offline

    14
    2u_{k+1} = 2(2k - u_k) = 4k - 2u_k, which is how the 4 arises. There's no need to rearrange for k.
    Offline

    2
    ReputationRep:
    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.
    • Thread Starter
    Offline

    0
    ReputationRep:
    (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.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Has a teacher ever helped you cheat?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Write a reply...
    Reply
    Hide
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.