Yes, the question came from the new Maths in Focus Ext 2 textbook in ch 5 Mathematical Induction Exercise 5.04 Q3 part b.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.
Would you be able to share a copy of that textbook, perhaps online through google drive? I know I'm late ahaha, but I'm really limited on resources.Yes, the question came from the new Maths in Focus Ext 2 textbook in ch 5 Mathematical Induction Exercise 5.04 Q3 part b.
Yes, I would have normally have solved this question with differentiation, but the textbook asked for the statement to be proved using Mathematical Induction, and was a "hence prove" follow on from part a.Heads up - using the method of differentiation (Which is significantly easier) and induction for such problems is quite common in a lot of 3 unit exams, so just make sure you are aware of that process
It's my understanding that the new syllabus Maths in Focus is better than the older ones written by Margaret Grove... but I agree that the old books were not good books.Maths in focus was never good. I haven't seen the book but I'm 99.9% sure it's trash like all the other ones.
The only thing it's 'sort of'? good for is explaining concepts from what I hear through other students.
It's ok, does some topics well, but has some material that is not covered by the others, which adds to the confusion of what to concentrate on. My pick is the new Cambridge text, followed by Fitzpatrick Ed 3 and the new Terry Lee.