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
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
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:
Subtract 504 from both sides:
and combine the numbers on the left to get:
Now factor 9 out of the right side:
For number 11, the basic definition of the Fibonacci numbers includes a rule saying we can obtain
by adding the previous two terms 
. For example, we have 
, or 
— basically, if we add two terms in a row, we will get the next one. So what happens if we add 
? For any natural number 
you choose, this expression ends up adding two terms in a row – for example, if 
, then 
will be 
– so it should equal the next term. What is the next term after 
? It’s 
. Thus 
.
Let me know if you have any followup questions. Good luck!
-Prof. Reitz