I just stumbled a "prove by induction" question and the answer says we should start with the RHS since it is easier to manipulate and I'm wondering if there is any techniques/Rules/ways to figure out which side is easier or likely to be easier to manipulate?
From this question I've taken that it is easier to manipulate a polynomial over a fraction but generally what other ways or rules are there?
$$\frac{x^n-1}{x-1} = 1 + x + x^2 + \dots + x^{n-1}$$
On
In order to prove $$\frac {x^n-1}{x-1} = 1+x+x^2+...+x^n$$ by induction, it is easier to start at RHS because it clearly shows that you only need to add x^[n+1} t0 both sides and use the induction hypotheses.
In general, it depends on the question.
There is no black and white rule that works all the time.
This may be very unsatisfying, but the answer (IMHO) to the question "How can I see which side is easier to manimulate?" is this: $$\mathbf{\text{Experience}}.$$
There is no general rule that tells you which side will be easier to manipulate. How could there be, when it is nearly impossible to formulate a good unambiguous definition of easy.