Re: HSC 2013 4U Marathon
by a telescoping sum.
Now expanding the binomial first instead of using a telescoping sum gives:
Now if we apply the summation, we sum each one individually, that is:
Now using our definition of
we obtain:
Using the symmetry property of binomial expansions gives:
Now since
and
we get:
Now consider the second part of the equation:
by definition.
Hence if we add the two parts together we obtain:
So the two terms on the end cancel out and give:
But this was equal to the value of the telescoping sum, so we can equate the two and get:
Taking everything from the RHS to the LHS except for the last term gives us the required result of:
Which can be re-written in the form asked:
Quod Erat Demonstrandum.