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