Question

The expansion of [2x² - (2/3x)]^h has 7 terms and one of them is 240ax^6. What's the value of a?
a) 2/9
b) 2/3
c) 4/9
d) 1/9

Can someone help me with this pls?

Answer 1

Are you familiar with the binomial theorem?

Answer 2

Kind of.

Answer 3

Using the binomial theorem you could expand it entirely, but because you aren't sure of what the exponent h is, you need to use the binomial theorem in a rather unique way. First, find where 240 is in pascal's triangle and how that corresponds to some a-choose-b

Answer 4

Although, when I look for it, it appears not to be there, so we'll have to be a little more clever.

Answer 5

This is very hard

Answer 6

I was making it slightly more difficult than it actually was.

Answer 7

First, since the second term in the binomial has a \(1\over x\), every term will have a different exponent for x. Thus, since it's 7 terms, you know \(h=6\).

Answer 8

If h=6 do I have to expand everything to get the answer?

Answer 9

Then, you need to find which term will have an \(x^6\). After a little bit of guess and check, you can find that the third term will have the \(x^6\).

Answer 10

So you only need to calculate\[\binom{6}{2} (2x)^4 \left({2\over{3x}}\right)^2\]

Answer 11

Mmmm I think I get it

Answer 12

That should be \((2x^2)^4\) not a \((2x)^4\)

Answer 13

If you calculate that out, you get \[{320\over3}x^6\] Which, after dividing by \(240\), you find your answer is \(4\over9\)

Answer 14

Why did you divide it by 240?

Answer 15

Because in the original question, it asked for what a was in \(240\;a\;x^6\) so you need to divide by 240 to get a.