Thread: 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!

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
I believe what you should be trying to prove is


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

to prove the formula for all n.

I hope this helps.

I'm just having trouble showing that,

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



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



The right hand side can then be rewritten as



Now using instead of for you should be able to complete the induction.

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.