Multiplying two binomials and forming conjectures (induction) (URGENT)

[Guest Aniruddh]

Hey Guys,

 

I have a Math test tomorrow so I need some help REAL quick.

 

When I was practicing binomials, I came there are several questions for finding the general term like (x-1)³((1/x) + x)⁶  Find the coefficient of x^-3 in the expansion. Our teacher told us that we have to expand both of the expressions manually and then multiply them to get the coefficient, but I refuse to believe that because that will take a lot of time and space and increase the probability of numbers as well. If anyone has a shorter way, please share.

 

On to induction, in our book there is this section called "Forming conjectures" in the induction chapter which explains how to find the nth term and the formula for sum of any series. The problem is that I didn't understand one bit of it - it's totally not clear and it has one example which use trigonometric identities which we haven't done yet so it all looks like nonsense to me. If someone could explain that to me, or redirect me to some resources that'd be great.

 

Our book has about 100 sums for both the chapters, and then I have all my past papers as well which will likely have another 50 or so excluding the questionbank. If someone could post a couple of the most difficult questions for both the topics (with solutions) that'd be really helpful as I will be saving a significant amount of time.

 

Thanks,

Aniruddh

Edited August 20, 2014 by Aniruddh