Tom's question at Yahoo! Answers regarding proof by induction

[MarkFL]

Here is the question:

Proove the following by mathematical induction?

while justifying your supporting arguments using the language of proof coherently, concisely and logically.
3 + 7 + 11 + 15 + ... to n terms = 2n^2 +n

Here is a link to the question:

Proove the following by mathematical induction? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[MarkFL]

Hello Tom,

The statement as given is not true, so I assume it is a typo, and instead the problem should be as follows.

We are given to prove by induction the following:

\(\displaystyle \sum_{i=1}^n(2i+1)=n^2+2n\)

Step 1: demonstrate the base case $P_1$ is true:

\(\displaystyle \sum_{i=1}^1(2k+1)=(1)^2+2(1)\)

\(\displaystyle 2(1)+1=1+2(1)\)

This is true.

Step 2: state the induction hypothesis $P_k$:

\(\displaystyle \sum_{i=1}^k(2k+1)=k^2+2k\)

Step 3: derive $P_{k+1}$ from $P_k$ to complete the proof by induction.

Let our inductive step be to add \(\displaystyle 2(k+1)+1\) to both sides of $P_k$:

\(\displaystyle \sum_{i=1}^k(2k+1)+2(k+1)+1=k^2+2k+2(k+1)+1\)

\(\displaystyle \sum_{i=1}^{k+1}(2k+1)=k^2+2k+1+2(k+1)\)

\(\displaystyle \sum_{i=1}^{k+1}(2k+1)=(k+1)^2+2(k+1)\)

We have derived $P_{k+1}$ from $P_{k}$ thereby completing the proof by induction.

To Tom or any other guests viewing this topic, I invite and encourage you to register and post other proof by induction problems either in our Pre-Calculus or Discrete Mathematics, Set Theory, and Logic forums depending on the nature of the problem, or course from which it is given.

Best Regards,

Mark.