Lreese Homework 2.pdf - Lathaniel Reese COT3100 Professor...

Lathaniel Reese
COT3100
Professor Bernd Owsnicki-Klewe
Homework 2
1) Prove: If n is a positive integer such that n mod 4 (the remainder of n under
division by 4) is 2 or 3, then n is not a perfect square.
With n = k^2, these four cases are considered;
1. If k mod(4) = 0, then k = 4q, for some integer q. Then, n = k 2 = 16 q 2 = 4(4 q 2 ) ,
which means n mod(4) = 0.
2. If k mod(4) = 1, then k = 4q + 1, for some integer q. Then, n = k 2 = 16 q 2 + 8 q + 1=
4(4 q 2 + 2 q) + 1, which means n mod(4) = 1.
3. If k mod(4) = 2, then k = 4q + 2, for some integer q. Then, n = k 2 = 16 q 2 + 16 q + 4 =
4(4 q 2 + 4 q + 1),which means n mod(4) = 0.
4. If k mod(4) = 3, then k = 4q + 3, for some integer q. Then, n = k
2
= 16 q
2
+ 24 q + 9 =
4(4 q
2
+ 6 q + 2) + 1,which means n mod(4) = 1.
If n is a perfect square, then n mod(4) is either 1 or 0. If it is not a perfect square,
then n mod(4) will be the other two possible options.

2) Consider the sum 3 + 6 + 9 + ... + 3n
a) Guess a formula P(n) for this sum, depending on n
b) Prove it by induction. [Base case: n = ? [2], P(k) [2], P(k+1) [3], P(k) →
P(k+1)
a) P(n) = 3n(n+1)/2
b) Base Case: n = 1
3 = 3
3n(n+1)/2 = 3(2)/2 = 3