I need some help with the questions shown in the attached .png file (especially 6-9). I have done some reading online relating to graphs and induction but I'm not sure how to prove these questions....

5. We have
P (n) =
n∑
i=1
3
4i
when n = 1, we have P (1) = 34 which is less than 1. That is P (1) < 1.
Suppose P (n− 1) < 1. Now we have to prove that P (n) < 1.
We have
P (n− 1) = 3
4
+
3
42
+ ... +
3
4n−1
Hence
1
4
P (n− 1) = 3
42
+ ... +
3
4n−1
+
3
4n
That is
1
4
P (n− 1) = P (n)− 3
4
this gives
P (n) =
3
4
+
1
4
P (n− 1) = 3 + P (n− 1)
4
As P (n− 1) < 1, hence 3 + P (n− 1) < 4, this gives 3+P (n−1)4 < 1. Hence
we have
P (n) < 1
Hence by induction, for all n, P (n) < 1.
7. We have given G is not connected. We claim: G′ is connected.
Let v and w are vertices. If vw is not an edge in G, then it is an edge in
G′, and so we have a...