Question

Prove by mathematical induction. helpp D:

Answer 1

Here's the claim\[3+5+7+\cdots+(2n+1)=n(n+2)\]
for \(n\ge1\).

Base case: Show that equality holds for \(n=1\):
\[2(1)+1=3=1(1+2)\]
Induction hypothesis: Assume equality holds for \(n=k\):
\[3+5+7+\cdots+(2k+1)=k(k+2)\]
Show that this implies equality for \(n=k+1\), i.e.
\[3+5+7+\cdots+(2(k+1)+1)=(k+1)(k+1+2)\]
So you understand what you have to do now, right?

Answer 2

\[\begin{align*}3+5+7+\cdots+(2(k+1)+1)&=(k+1)(k+1+2)\\
3+5+7+\cdots+(2k+3)&=k^2+2k+1+2k+2\\
\color{red}{3+5+7+\cdots+(2k+1)}+(2k+2)+(2k+3)&=k^2+4k+3
\end{align*}\]

Answer 3

thanks!