Using the Binomial Theorem, find the first 5 terms in the expansion and estimate

In summary, the conversation is discussing how to find an estimate for the value of 1.07^9 using the Binomial Theorem. The speaker added the first 5 terms of the expansion and wrote that as their estimate. They were unsure if this was all the question was asking or if there was a further mathematical step involved. However, another speaker clarified that adding the first 5 terms is a mathematical approach and that the remaining terms can be neglected in the estimation.
  • #1
singleton
121
0
Okay, well, perhaps I am already done, perhaps not. This is why I seek your wisdom :)

If you are asked to find the first 5 terms of the expansion (1 + 0.07)^9 using the Binomial Theorem, and then asked to use these terms to estimate the value of 1.07^9 what would you put?

I added the 5 terms and wrote that I estimate the value of 1.07^9 would be that sum.

Is this all the question is asking? Or are you supposed to go further with the terms and do something mathematical? :confused: :yuck:
 
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  • #2
Yeah, you added the first 5 terms and that was your estimate of 1.07^9. That seems like all there is.
 
  • #3
singleton said:
Okay, well, perhaps I am already done, perhaps not. This is why I seek your wisdom :)

If you are asked to find the first 5 terms of the expansion (1 + 0.07)^9 using the Binomial Theorem, and then asked to use these terms to estimate the value of 1.07^9 what would you put?

I added the 5 terms and wrote that I estimate the value of 1.07^9 would be that sum.

Is this all the question is asking? Or are you supposed to go further with the terms and do something mathematical? :confused: :yuck:

Why would writing out the first 5 terms of the binomial expansion and adding them not be "mathematical"?
 
  • #4
Oh, well of course that was a mathematical approach...

But I thought this problem would require more thought/something I don't know, as the work it is along with is more than this simple. I thought that I was being tricked :redface:

I guess sometimes a cigar is just a cigar, eh? :D
 
Last edited:
  • #5
you can find the first five terms by expension.they asked to find the estimate value,which can be obtained by adding the first 4 to 5 terms.the others terms are so small that they can be neglected...
 

1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula that allows us to expand binomials, which are expressions with two terms, raised to a certain power. It is written as (a + b)^n and can be used to find the coefficients of each term in the expansion.

2. How do you use the Binomial Theorem to find the first 5 terms in the expansion?

To find the first 5 terms in the expansion, we use the formula (a + b)^n = ∑(n choose k) * a^(n-k) * b^k, where n is the power, k is the term number (starting from 0), and ∑ represents the sum of all terms. This formula allows us to calculate the coefficients for each term and plug them into the expansion.

3. Can you provide an example of using the Binomial Theorem to find the first 5 terms in the expansion?

Sure. Let's say we want to expand (x + y)^4. Using the formula, we have (4 choose 0) * x^4 + (4 choose 1) * x^3 * y + (4 choose 2) * x^2 * y^2 + (4 choose 3) * x * y^3 + (4 choose 4) * y^4. Simplifying this, we get x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. These are the first 5 terms in the expansion.

4. Why is it important to estimate the terms in the expansion?

Estimating the terms in the expansion allows us to get a rough idea of the overall value of the expression without having to calculate every single term. This is especially useful when dealing with large powers, as it can save us time and effort in calculating the entire expansion.

5. Are there any real-world applications of the Binomial Theorem?

Yes, there are many real-world applications of the Binomial Theorem. It is commonly used in statistics and probability to calculate the likelihood of certain outcomes. It is also used in finance and economics to calculate compound interest and binomial option pricing. Additionally, it has applications in engineering, physics, and other fields that involve calculations with binomial expressions.

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