Question

Find the specified nth term in the expansion of the binomial. (Write the expansion in descending powers of x.)
(5x + 4y)6, n = 4

Answer 1

what subject is this in? algebra or calculus?

Answer 2

precalculus

Answer 3

hmmm it's been awhile on this for me, so give me a minute

Answer 4

is that supposed to be to the 6th power?

Answer 5

yeah it is sorry

Answer 6

no problem. First thing to think about is the coefficients. They come from Pascal's triangle (and I feel like there's somewhere else too, but I don't know where). The sixth row of the triangle is "1,5,10,10,5,1" so these are the coefficients on your expansion

Answer 7

so now for the rest. \[(a+b)^n=a^n + 5a^{(n-1)}b + 10a^{n-2}b^2 ... b^n\]
I just threw in the coefficients for the 6th power, but they are different for other powers. All you have to do is start with a to the n (whatever exponent you've got) and b to the 0, then in the following terms you just keep adding one to the exponent on b and taking one from the exponent on a until you end up with a^0 and b^n. Just keep in mind that, in this case, a=(5x) and b=(4y), so keep those terms together as you go.

Answer 8

I've found the answer, so if you want you can type me what you get to make sure it's right :)

Answer 9

ok hold on a sec

Answer 10

i got 360x^2y^4 but for some reason I think thats wrong?

Answer 11

ok sorry I just realized I used n to mean the original exponent and they use n to mean the 4th term in the sequence. That might have been confusing. Lets start at the beginning, the first term in the sequence or the n=1 term is:
\[(5x)^6\]
with me?

Answer 12

ok yeah

Answer 13

so the second term is \[5(5x)^5(4y)\]
because we're at position 2 in the series, or at n=1... and the coefficient that corresponds with that position in our series is 5 from the triangle

Answer 14

okay

Answer 15

whoops I meant n=2 in that last one. We're at position 2.

Answer 16

ok so now position 3 or at n=3
\[10(5x)^4(4y)^2\]

Answer 17

ok

Answer 18

so keeping in mind that the coefficients are 1,5,10,10,5,1... can you find the next one, n=4?

Answer 19

10(5x)^3(4y)^3?

Answer 20

Boom! you got it!

Answer 21

Do you feel like you have it? Or do you want to do another one like it?

Answer 22

no i think I got it! thanks for your help!

Answer 23

you bet!