How to Simplify (11-1)^9 Using Binomial Expansion?

In summary, the conversation discusses a binomial expansion problem involving the expression (11-1)^9. The answer is found by expanding the expression and simplifying it to (9 choose 0)11^9 + (9 choose 1)11^8*(-1) + (9 choose 2)11^7*(-1)^2 and so on. The individuals in the conversation also discuss the values of a, b, and n in the binomial expansion formula. Ultimately, the problem is solved and the asker thanks the other person for their help.
  • #1
mr_coffee
1,629
1
Hello everyone.

I'm studying for my exam and I'm reveiwing some problems but this one isn't making sense to me:

11^9(9 choose 0) - 11^8(9 choose 1) + 11^7(9 choose 2) - ... - 11^2(9 choose 7) + 11^1 (9 choose 8) - 11^0 (9 choose 9)


answer:
(11-1)^9 = 10^9 = 1,000,000,000.

work looked like:
(9 choose 0) 11^9 (-1)^0 + (9 choose 1)11^8(-1)^1


Can someone explain to me how they did this?
I see the 11 is decreasing, and the signs are alternating, it looks like a binomial expansion but I'm not seeing how they simplified that.

Thanks
 
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  • #2
It IS a binomial expansion. It's 11k*(-1)9-k*9choosek with k going from 0 to 9

Try expanding (11-1)9 and compare them
 
  • #3
col_alg_tut54binomialthe.gif


So would
a = 11
and
b = -1
and
n = 9

(11-1)^9 = (9 choose 0)11^9 + (9 choose 1)11^8*(-1) +(9 choose 2)11^7*(-1)^2...

I see this is going to work, from the written out expansion, i can see n = 9, b = -1, a = 11,


ahh i got it@!

thank u!
 

1. What is a geometric progression?

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, called the common ratio. For example, in the sequence 1, 2, 4, 8, 16, the common ratio is 2.

2. How do I determine if a sequence is a geometric progression?

To determine if a sequence is a geometric progression, you can check if the ratio between any two consecutive terms is constant. If it is, then the sequence is a geometric progression. Another way is to see if the sequence can be written in the form a, ar, ar^2, ar^3, ... where a is the first term and r is the common ratio.

3. What is binomial expansion?

Binomial expansion is a mathematical process for expanding binomial expressions, which are expressions with two terms connected by addition or subtraction. It involves using the binomial theorem to find the coefficients of each term in the expansion.

4. How is binomial expansion related to geometric progression?

Binomial expansion can be used to find the terms in a geometric progression. For example, if you have an expression (a + b)^n, you can expand it using binomial expansion to get the terms a^n, a^(n-1)b, a^(n-2)b^2, ..., b^n. This is similar to the terms a, ar, ar^2, ..., ar^(n-1) in a geometric progression.

5. Can binomial expansion be used to find large values?

Yes, binomial expansion can be used to find large values. For example, if you have an expression (a + b)^n, you can expand it using binomial expansion to find the value of (a + b)^n for large values of n, such as 10^9.

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