1. Induction Help

∑ i=1 to n, √[1+(1/i^2)+(1/(1+i^2))] = n(n+2)/n+1

First I did the base case of p(1) showing 3/2 on the LHS equals the 3/2 on the RHS.
Then I assumed p(k) and wrote out the formula with k in it.
Then prove p(k+1)= p(k)+ √[1+1/(k+1)^2+1/(k+2)^2]
=k(k+2)/k+1 + √[1+1/(k+1)^2+1/(k+2)^2]
Then I squared each to get rid of the square root.
(k(k+2)/(k+1))^2+ (k+1)^2/(k+1)^2 + 1/(k+1)^2 + 1/(k+2)^2
Now I'm stuck any Guidance would be great thanks!

2. Re: Induction Help

There are a couple of problems with your post. Firstly I don't believe that formula is correct. According to your formula the LHS when n=1 is $\displaystyle \sqrt{1+1+\frac{1}{2}}=\frac{\sqrt{10}}{2} \neq \frac{3}{2}.$
I believe what you should be trying to prove is
$\displaystyle \sum_{i=1}^{n}\sqrt{1+\frac{1}{i^2}+\frac{1}{(1+i) ^2}}=\frac{n(n+2)}{n+1}.$

Secondly your approach for taking the inductive step seems a little strange- in such proofs you prove the base case (i.e. n=1) and then assume the formula to be true for n=1,..,k. Then you consider the LHS when n=k+1, and use the fact that
$\displaystyle \sum_{i=1}^{k+1}f(i)=\sum_{i=1}^kf(i)+f(k+1)$
to prove the formula for all n.

I hope this helps.

3. Re: Induction Help

I'm just having trouble showing that, $\displaystyle \frac{k(k+2)}{k+1} + \sqrt{1+\frac{1}{k+1^2}+\frac{1}{(1+k+1)^2}}=\frac {k+1(k+1+2)}{k+1+1}$

4. Re: Induction Help

You've got to mess around a bit with the algebra:

$\displaystyle \sqrt{1+\frac{1}{i^2}+\frac{1}{(i+1)^2}}=\frac{1}{ i(i+1)}\sqrt{i^2(i+1)^2+(i+1)^2+i^2},$

Multiplying out the term in the square root and collecting like terms gives

$\displaystyle \frac{1}{i(i+1)}\sqrt{i^4+2i^3+3i^2+2i+1}=\frac{1} {i(i+1)}\sqrt{(i^2+i+1)^2}=\frac{i^2+i+1}{i(i+1)}$

The right hand side can then be rewritten as

$\displaystyle g(i)=\frac{i^2(i+2)-(i-1)(i+1)^2}{i(i+1)}.$

Now using $\displaystyle g(i)$ instead of $\displaystyle \sqrt{1+\frac{1}{i^2}+\frac{1}{(i+1)^2}}$ for $\displaystyle i=k+1$ you should be able to complete the induction.

5. Re: Induction Help

You might want to take a look at this topic:

Sum of a finite series.

for a way to simplify your induction hypothesis.

edit: If I had taken the time to actually read the post above this one before posting, I would have seen the same simplification has already been given.