PreCalculus question

Question

Use the Binomial Theorem to find the coefficient of the fifth term: (2x + 1)^7

a. C(7,3)

b. C(7,4)

c. C(6,5)

c. C(7,5)

Answer 1

Hi Clarissa! If you are a little confused on binomial theorem, here is the link a video that might help: https://www.youtube.com/watch?v=iPwrDWQ7hPc.

The binomial theorem states that (a+b)ⁿ = ∑C(n,k)*a^n-kb^k. In your case, you are looking at (2x+1)⁷. So in this problem specifically, '"a" = 2x, "b" = 1, "n" = 7, and k = 4. The value of k is 4 and not 5, since k gives you the power of the term. The fifth term will be to the fourth power since the first term is a constant.

Try plugging in these given values into C(n,k)*a^n-kb^k .Whatever constant you have will be your answer. One thing I might note is that the answer differs depending if the question is asking for the fifth term of "x" or "2x". Considering your options, it seems to be expanding the series upon "2x". Give it a try and let me know what you get (I got "b", assuming expansion upon 2x). Let me know if you have any questions!

Answer 2

Hey Clarissa,

To answer this problem, first recall the binomial theorem: (a+b)ⁿ = ∑ⁿ_k=0 (_kⁿ) a^n-kb^k

(apologies for the wonky formatting- you can cross-reference this format in your textbook, study materials, or online to see exactly where the radicals and variables are placed).

Fortunately we don't have to contend with the whole formula- we are just focusing on this part: (ⁿ_k).

This can also be written C(n, k) as it is in your problem, and is read as "n choose k." It is equivalent to n!/k!(n-k)!

So, now that we've established the steps, what are n & k? Let's look at the expression given, (2x+1)⁷

This expression is already in the format (a+b)ⁿ = (2x+1)⁷, so n=7. Great! This means we can eliminate option C.

What about k?

Recall that we always start with k=0, as noted under sigma ∑. So, for the 1st term k=0. We increase k by 1 as we move along the terms. For the 2nd term k=1, for the 3rd term k=2, etc. Note that k is always 1 less than the term number. So for the 5th term, k=4.

Knowing that C(n, k), n=7 & k=4, our final answer is b. C(7,4).