I’m not too sure exactly how to approach this question. Would anyone be able to give me any helpful advice or some sort of direction? I have a little problem with induction.
Prove by induction that:
$$1 + 3 + 5 + 7 +…+ (2n+1) = (n+1)^2 \quad\forall n \in \mathbb N $$
Let $P(n)$ be the statement $1 + 3 + 5 + \cdots + (2n + 1) = (n + 1)^2$.
The first thing you need to do is to show that $P(1)$ is true. In this case that means showing that $$1 + (2 \cdot 1 + 1) = (1 + 1)^2$$
Now that you have established that the proposition $P(n)$ is true for some integer $m$ (since it is true for $n = 1$), you may assume that $P(m)$ holds for some $m \in \mathbb{N}$.
The final step is to show that $P(m) \Rightarrow P(m + 1)$. Since $P(1)$ holds, establishing that $P(m) \Rightarrow P(m + 1)$ demonstrates that $P(n)$ holds for each $n \in \mathbb{N}$ since $P(1) \Rightarrow P(2) \Rightarrow P(3) \Rightarrow \cdots$.
In this case, to prove $P(m + 1)$ holds, you must show that
$$1 + 3 + 5 + \cdots + (2m + 1) + [2(m + 1) + 1] = [(m + 1) + 1]^2$$
under the assumption that $P(m)$ holds, that is, under the assumption that
$$1 + 3 + 5 + \cdots + (2m + 1) = (m + 1)^2$$
For $n=1$ we have that $1+3=(1+1)^2=4$. Suppose that the relation is true for $n=m$ then
$$1+3+\cdots +(2m+1)=(m+1)^2$$
then
$$1+3+\cdots +(2m+1)+(2(m+1)+1)\\=(m+1)^2+(2(m+1)+1)\\=((m+1)+1)^2=(m+2)^2$$
You should show that if $X = \lbrace n\in \mathbb{N}; 1 + 3 + … + (2n+1) = (n+1)^2\rbrace \subset \mathbb{N}$ is such that $0 \in X$ and if $n \in X \Rightarrow n+1 \in X$ then $X=\mathbb{N}$.
Now, for $n=0$ we have
$$1 = 1^2 = 1$$
Suppose that for $n=k$ the assertion is valid. We must then prove that it holds for $n=k+1$. In fact
$$\begin{align} 1 + 3 + … + (2k+1) + (2(k+1) + 1) = & (k+1)^2 + 2(k+1) + 1\\ = & k^2 + 2k + 1 + 2k + 2 + 1 = k^2 + 4k + 4 = (k+2)^2\end{align}$$
Then $n=k+1 \in X$.
By induction we have that it is true for all $n \in \mathbb{N}$.
As an exercise of it try to show that
i. $n \geq 4 \Rightarrow n! > 2^n$
ii. $2(1 +2 + …+ n) = n(n+1)$
iii. $(a-1)(1+a+…+a^n) = a^{n+1} -1$