Proofs by Induction: are my two proofs correct? I have been trying to understand how proof by mathematical induction works, and I am struggling a bit. But, I think I am understanding it and I just want to verify that what I am doing is correct (and if not, why?)
I have attached a screenshot (as a link) of my problem (black ink) and my work (red ink). My main issue is understanding what the final conclusion should be. What I did was check to see if the left and right side of the problem were equal after assuming $k + 1$ is true, and adding the appropriate terms to both sides, and simplifying.
So, in my final steps of the induction phase, my question is, did I reach the right result?
Prove: $1 + 3 + 6 + \cdots + \dfrac{n(n + 1)}{2} = \dfrac{n(n + 1)(n + 2)}{6}$.
Base:  $P(1) = 1$.
Induction:
\begin{align*}
\underbrace{1 + 3 + 6 + \cdots + \frac{k(k + 1)}{2}}_{\dfrac{k(k + 1)(k + 2)}{6}} + \frac{(k + 1)(k + 2)}{2} & = \frac{(k + 1)(k + 2)(k + 3)}{6}\\
\frac{k(k + 1)(k + 2)}{6} + \frac{(k + 1)(k + 2)}{2} & = \frac{(k + 1)(k + 2)(k + 3)}{6}\\
\frac{k(k + 1)(k + 2) + 3(k + 1)(k + 2)}{6} & = \frac{(k + 1)(k + 2)(k + 3)}{6}\\
\frac{(k + 1)(k + 2)(k + 3)}{6} & = \frac{(k + 1)(k + 2)(k + 3)}{6}
\end{align*}
Prove: $5 + 10 + 15 + \cdots + 5n = \dfrac{5n(n + 1)}{2}$
Base:  $P(1) = 5$
Induction:
\begin{align*}
5 + 10 + 15 + \cdots + 5k + 5(k + 1) & = \frac{5k(k + 1)}{2} + 5(k + 1)\\
\frac{5k(k + 1)}{2} + 5(k + 1) & = \frac{5k(k + 1)}{2} + 5(k + 1)
\end{align*}
My problem and my work

In a proof by mathematical induction, we wish to establish that some property $P(n)$ holds for each positive integer $n$ (or for each integer greater than some fixed integer $n_0$).  We must first establish that the base case holds.  Once we establish that it holds, we may assume the property holds for some positive integer $k$.  We then need to prove that if $P(k)$ holds, then $P(k + 1)$ holds.  Then, if our base case is $P(1)$, we obtain the chain of implications
$$P(1) \implies P(2) \implies P(3) \implies \cdots$$
and $P(1)$, which establishes that the property holds for every positive integer.
You should not assume $P(k + 1)$ is true.  We must prove that $P(1)$ holds and that if $P(k)$ holds, then $P(k + 1)$ holds for each positive integer $k$.
Let's look at the first proposition.
Proof.  Let $P(n)$ be the statement that
$$1 + 3 + 6 + \cdots + \frac{n(n + 1)}{2} = \frac{n(n + 1)(n + 2)}{6}$$
Let $n = 1$. Then
$$\frac{n(n + 1)}{2} = \frac{1(1 + 1)}{2} =\frac{1 \cdot 2}{2} = 1 = \frac{1 \cdot 2 \cdot 3}{6} = \frac{1(1 + 1)(1 + 2)}{6}$$
Hence, $P(1)$ holds.
Since $P(1)$ holds, we may assume $P(k)$ holds for some positive integer $k$.  Hence,
$$1 + 3 + 6 + \cdots + \frac{k(k + 1)}{2} = \frac{k(k + 1)(k + 2)}{6}$$
This is our induction hypothesis.
Let $n = k + 1$.  Then
\begin{align*}
1  + 3 + 6 + & \cdots + \frac{k(k + 1)}{2} + \frac{(k + 1)(k + 2)}{2}\\ 
& = \frac{k(k + 1)(k + 2)}{6} + \frac{(k + 1)(k + 2)}{2} && \text{by the induction hypothesis}\\
& = \frac{k(k + 1)(k + 2) + 3(k + 1)(k + 2)}{6}\\
& = \frac{(k + 1)(k + 2)(k + 3)}{6}\\
& = \frac{(k + 1)[(k + 1) + 1][(k + 1) + 2]}{6}
\end{align*}
Thus, $P(k) \implies P(k + 1)$ for each positive integer $k$.
Since $P(1)$ holds and $P(k) \implies P(k + 1)$ for each positive integer $k$, $P(n)$ holds for each positive integer $n$.$\blacksquare$
I will leave the second proof to you.
