if you get to the expression it should be equal to, can you do the induction?

this is a telescoping sum. meaning, you can write it in a different form that when you expand it, most of the terms will cancel out, leaving you with what the sum actually is.

note that (if you can't expand like that using algebra, just use partial fractions)

so that

now expand the sum on the right to see what cancels out. that will get you your formula to do the induction with