Unless
required to prove this by induction, this is an excellent example of where an alternative method is much easier.
This result is the direct application of differentiation to the binomial theorem.
The binomial theorem states that
Differentiating with respect to
x yields:
Noting the term in the sum is zero when
, we have:
as required
Something to bear in mind here: Questions sometimes use the word "otherwise", as in "Using induction or otherwise, prove ...". The word "otherwise" pretty much always indicates that
either there is an alternative method that is much quicker,
or that there is an alternative approach that will lead to a huge mess and you should avoid it. A question with a preferable "otherwise" approach offers the chance to pick up time in an exam, which you can then use on other questions. It is worth stopping to consider when the word "otherwise" appears and asking if a better way is available.