Term Test #1

i find Term Test #1 to be fair, i knew the answers to all the questions unfortunately, in the 50 mins period span, i managed to only write a structure and a few of the cases for #3 (which was the subset question) without finish proofing it.

Question 1 was pretty straight forward, using simple induction and proof 3^n > n^3 + 10, basically it was algebra and inequality manipulation, going from 3^(n+1) to showing its greater than (n+1)^3 + 10.

In Question 2, although some people i talked to used complete induction, i just used simple, it was like the sum of the fib numbers, we just need to proof that f(n) > 2n, with this, i first say f(n+1) = f(n-1) + f(n), then by using the def of f(n) i rearranged so that f(n-1) = f(n) - f(n-2), such that its 2f(n) - f(n-2), and then everything works itself from there using the I.H. With a bit manipulation we arrive to 2(n+1) which is what we wanted.

In question 3, it was number of subsets such that its 3*2^n-2 if 1 or 2 or both are omitted. It was a fairly simple proof using cases, split it into 1 being omitted, 2 being omitted, then both elements being omitted, and soon enough adding them up, and taking away elements that were repeated(such as empty case), we would be able to arrive at our conclusion. Unfortunately i spent some time thinking about the inequalities manipulation in question 1, and thus not had time to finish question3, however i did put a conclusion in even though i did not finish the proof, as i see the marking scheme from the past term test that marks were taken off if certain things were omitted. funny how i mention omitted things and question 3 has the omit elements.

maybe i can proof using induction my test mark if the n required things were omitted eh?