Just try to use the common denominator:
which is exactly the RHS.
But the modification of the RHS is perhaps not so important.
Just try to use:
or (the same equation with n+1 instead of n
in the inductive step of your proof.
EDIT: BTW the sums of this type are called
telescopic. With a little practice you can recognize easily at least the simple ones - and then you see the result immediately (that was the reason why I rewrote the RHS - since I saw that this expression is what I get from the telescopic sum.