Can anyone help me with this induction proof?

Question

I need to prove that:

$$1+3+5+\cdots+(2n+1)=(n+1)^2$$

I first verify that it is true for $n=0$ and then I check for $n=k$

If it is true for $n=k$, it should be true for $n=k+1$

Therefore:

$$1+3+5+\cdots+(2k+1)+(2(k+1)+1)=(k+1)^{2}+(2(k+1)+1)$$ should be equal to $$((k+1)+1)^2$$

I just can't figure out how to show that $$(k+1)^2+(2(k+1)+1)=((k+1)+1)^2$$

Can anyone help me out here?

Answer 1

$$(k+1)^2+2(k+1)+1=k^2+2k+1+2k+3=k^2+4k+4=(k+2)^2$$

Answer 2

By using the Binomial Theorem $$\begin{align} \text{RHS}=((k+1)+1)^2=(k+1)^2+2\cdot1\cdot(k+1)+1&=(k+1)^2+2(k+1)+1\\ &=(k+1)^2+(2(k+1)+1)=\text{LHS} \end{align}$$