I'm doing a question on strong induction and sometimes I have a hard time understanding where to take my proofs, I will detail the questions and my answers to them.
The rules to the game are as follows:
Begin with a pile of n stones and a score of 0 points. In the first move, split the pile into two possibly unequal sub-piles, multiply the number of stones in one sub-pile by the number of stones in the other sub-pile and add that product to the score. Continue by successively splitting each newly created pile of stones that has at least two stones into a pair of sub-piles which are not necessarily the same size. Each time you split a pile into two sub-piles, multiply the number of stones in one of these sub-pile by the number of stones in the other sub-pile and add that product to the score. Continue the game until there are n piles of 1 stone each. The score at that point is the final score for the game.
we define $G(n)$ to be the score, the following is the conjecture in symbolic form
$$\forall n \in \mathbb N^+ G(n)=n(n-1)/2$$
a) Your conjecture has laready been stated in symbolic form: it is a statement of the from $\forall n \in \mathbb N^+, P(n)$. What is the predicate function $P(n)$
So I'm pretty sure $P(n)$ in this case is just $P(n)=n(n-1)/2$ but I am not 100% sure as it's just seems too obvious, I feel as though I'm missing something here
b)Base Case:
$$P(1)=1(1-1)/2$$ $$P(1)=1(0)/2$$ $$P(1)=0/2$$ $$P(1)=0$$
Base Case holds
My understanding of strong induction comes from a handout in class which states the following:
Let $P(n)$ be a predicate defined for natural numbers $n$. Let $a$ and $b$ be fixed natural numbers s.t. $a \le b$.
and the base case is defined as the following: $P(a), P(a+1), ..., P(b)$ are all true, however in my case I can't seem to figure out what my $b$ is supposed to be here, my $a$ is my starting point at $1$ if I'm understanding correctly.
c)Inductive setup:
let $n=k$ s.t $P(k)=k(k-1)/2$ be true, this is my inductive hypothesis (at least I think)
$P(k+1)= k(k+1)/2$ for $\forall k \in \mathbb N^+ s.t. k \ge 1$
d) Inductive step
This is as far as I've gotten, because I can't seem to understand the whole $P(a), P(a+1), ..., P(b)$ I don't really know where to go from here. i appreciate any help I can get, I prefer to not be given the answer but rather shown the correct path since I am gonna have to do more of these later on and I would like to learn as much from this as possible, thank you.
I have also seen this: Making sense of word problem which is very very similar but unfortunately after reading through it, it has not helped me get a better understanding of the question.