Chapter 1 · Algebraic Proofs · Concept 1
1-3 Applications of the Binomial Theorem
So far you have used the binomial theorem to find a coefficient. Now you will use it to prove that something is true.
Guiding question
If you know what (1 + x)n opens out to, what can you learn by putting a number in for x?
Learning outcome. Employs the binomial theorem in mathematical and computational applications.
A question here does not ask you to find a number. It gives you a long line of nCr terms and says prove this is true.
That sounds much harder than it is. Every single one is the same two moves:
Move 1
Write out (1 + x)n with the binomial theorem.
This line is the same every time. You never change it.
Move 2
Put the right number in for x.
This is the only part that changes. It is the whole lesson.
The line you always start from
(1 + x)n = nC0 + nC1x + nC2x2 + …… + nCnxn
Then put a suitable number in for x to get the equality you were asked for.
Why 1 + x and not something else? Because 1 to any power is still 1. So the first part disappears from every term and only x is left to work with.
Look at the numbers sitting in front of the nCr terms. They are the powers of x, so they tell you what x is.
| What you see in front | So x is | And the answer is (1 + x)ⁿ |
|---|---|---|
| 1, 1, 1, 1, … (nothing written) | 1 | 2ⁿ |
| 1, −1, 1, −1, … (signs swap) | −1 | 0 |
| 1, 2, 2², 2³, … | 2 | 3ⁿ |
| 1, −2, 2², −2³, … | −2 | (−1)ⁿ |
| 1, ½, ¼, ⅛, … with swapping signs | −1/2 | (1/2)ⁿ |
Every question on this page is one of these five rows. Learn to read the row and you are done.
Pick a value for x and add the terms up yourself. The long way and the short way always land on the same number. That is what a proof of this kind is saying.
Put 1 in place of x, one term at a time. Watch the running total.
running total after 1 term: 1
The short way: (1 + 1)5 = 32— which is 2ⁿ with n = 5
every term is added
Prove that nC0 + nC1 + nC2 + …… + nCn = 2n
- 1
Look at what is in front of each term.
1, 1, 1, …… — there is nothing written in front of any of themRow one of the table above. Nothing in front means x = 1.
Work with a partner. Do this one together, then talk about how you knew what x was. Send it to your notebook first — it is easier to work on paper.
Prove that nC0 − nC1 + nC2 − …… + (−1)n nCn = 0
- 1
Read the signs.
+ − + − ……They swap every time. Row two of the table: that means x = −1.
Prove that these equalities hold.
By using the binomial theorem, prove that (8n − 1) is divisible by 7, where n is a positive integer.
Substitute on BOTH sides.
If you put x = 1 into the right side you must put it into (1 + x)ⁿ as well. Changing one side only is not an equation any more, and it is the commonest way to lose every mark on the question.
Signs that swap mean a negative x.
+ − + − comes from x = −1. + then powers of 2 with swapping signs comes from x = −2. Read the signs before you write anything.
Dividing by 2 is multiplying by 1/2.
Exercise 3 has nCr ⁄ 2ʳ, which looks nothing like a power of x until you see it as (1/2)ʳ. With the swapping signs that makes x = −1/2.
n must be positive.
The Try proof ends with 0ⁿ = 0, and that is only true when n is 1 or more. At n = 0 the sum is 1, not 0 — which is exactly why the question says n is a positive integer.
That is Concept 1 finished. The review quiz covers all three lessons.
Keep reading — this lesson marks itself done automatically.