induction

Been trying for ages - atp I just want to see it solved. Please help me lol.

I have come across some questions that are asking me to prove the general formula for a certain sequence. I do now know how to tackle these questions. Can someone please tell me how to do them? Here is a problem from the question set that appeared: A sequence is given by the first order...

For the following question when I am trying to prove the n = k+1 case am I allowed to substitute it in the first line of the question and also in the second line of the question? What I am trying to say is am i allowed to assume:

hey, just encountered an issue with induction when testing the base case n=2 given n>=2 i thought it was only if n=1 we test the first value 1/2^2, but since n>=2, we test n=2 which includes the first 2 terms 1/2^2 + 1/3^2 but this isn't the case with HSC solutions, eddie woo includes the...

hey users, just got a question about strong induction for the assumption case, what would be the condition to assume n=<k or assume n=k, and what is the difference and would we lose marks for using wrong one? also, would we ever need to assume/prove n=<k+1 thanks

hey guys, can anyone solve this hard Q? thanks in advance

Hi all, Need help with 'strong' induction, a few questions on this topic in cambridge but they don't provide examples so im not sure how it differs from normal induction (n=1, n=k, n=k+1). Any help is appreciated. Example below: