Thread: How do I solve this expansion?

I have two questions which I don't know how to solve... I really need any help you can give. They're both on expansions, which I missed due to being absent for a week. I would love it if you could give me the answer, and explain how you got it. Thanks

65. This is a question about the binomial expansion.

You need to know that

(a+b)^n = sum_{r=0 to n} [n!/((n-r)! r!] a^{n-r} b^r

so in your case a=x^3, b=-y^2, n=20, and as you are asked for the 8-th
term r=7.

So the 8-th term is:

[20!/(13! 7!)] (x^3)^13 (-y^2)^7

which is answer D.

RonL

68. This is again a binomial expansion problem. So again we are interested in:

(a+b)^n = sum_{r=0 to n} [n!/((n-r)! r!] a^{n-r} b^r

but in this case a=1, b=2x and n=6, and we are interested in the first four terms:

6!/(6! 0!) 1^6 (2x)^0 + 6!/(5! 1!) 1^5 (2x)^1 + 6!/(4! 2!) 1^4 (2x)^2 + 6!/(3! 3!) 1^3 (2x)^3 = 1 + 6 (2x) + 15 (2x)^2 + 20 (2x)^3

......................= 1 + 12x + 60x^2 + 160x^3

Which is answer J.

RonL