Exam Review 3 UPDATE

UDPATE 2 (11/5): The proof of #10 had a typo.
Near the end, after the line “Subtracting 504 from both sides”, the right side should end with a “-504” rather than “+504”.  Similarly, in the next line, the final parentheses should end with “-56” rather than “+56”.

UPDATE: The answer key has been added to the review sheet – let me know if you find an error!

Hi everyone,

Be aware, there was typo in problem number #9.  The left side of the equation should end in “n(n+2)” instead of “n(n+1)”.  Apologies for this error!

-Prof. Reitz

2 responses to “Exam Review 3 UPDATE

  1. Hello,
    my first question is on number 10.
    I don’t understand how from:
    4^(3(k+1)) + 512 = 576a you get to 4^(3(k+1) )+ 8 = 9(64a+56)
    I know that we should get to 4^(3k) + 8 = 9a
    And my second question is on number 11:
    How do you get:
    F2k + F2k+1 = F2k+2 ???
    Thanks beforehand

  2. Hi Albina,

    Thanks for writing – here goes:

    In number 10, I’m not sure if it helps but I had a typo – it should read “-56” instead of “+56”. My steps are as follows.
    Start with: 4^{3(k+1)}+512=576a
    Subtract 504 from both sides: 4^{3(k+1)}+512 - 504=576a -504
    and combine the numbers on the left to get: 4^{3(k+1)}+8=576a-504
    Now factor 9 out of the right side: 4^{3(k+1)}+8=9(64a-56).

    For number 11, the basic definition of the Fibonacci numbers includes a rule saying we can obtain F_n by adding the previous two terms F_{n-2}+F_{n-1}=F_n. For example, we have F_{12}+F_{13}=F_{14}, or F_{35}+F_{36}=F_{37} — basically, if we add two terms in a row, we will get the next one. So what happens if we add F_{2k} + F_{2k+1}? For any natural number k you choose, this expression ends up adding two terms in a row – for example, if k=3, then F_{2k} + F_{2k+1} will be F_6+F_7 – so it should equal the next term. What is the next term after F_{2k} + F_{2k+1}? It’s F_{2k+2}. Thus F_{2k} + F_{2k+1}=F_{2k+2}.

    Let me know if you have any followup questions. Good luck!
    -Prof. Reitz

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