Binomial theprem and expansion

In summary, the conversation is about finding the last few terms of the expression (1+x)^n and using the binomial coefficients to prove a given inequality. The last few terms are n(n-1)xn-2/2!, nxn-1/1!, and xn. The conversation also includes a request for guidance on how to use the expression to prove an inequality.
  • #1
rohan03
56
0
(1+x)^n=1+nx/1!+(n(n-1) x^2)/2!+⋯+ what are the last few terms of this ? I looked and tried but don't seem to get any textbook answer for this.
 
Mathematics news on Phys.org
  • #2
hi rohan03! :smile:

(try using the X2 and X2 buttons just above the Reply box :wink:)

the binomial coefficients are symmetric (nCr = nCn-r),

so it ends … + n(n-1) xn-2/2! + nxn-1/1! + xn :wink:
 
  • #3
Thank you . Can I prove with the help of this :
(1+n)n ≥ 5/2nn- 1/2n n-1
 
  • #4
and if yes - please guide on how
 
  • #5
rohan03 said:
Thank you . Can I prove with the help of this :
(1+n)n ≥ 5/2nn- 1/2n n-1

just write out the last three terms …

what do you get? :smile:
 
  • #6
This was also posted under "homework" so I am closing this thread.
 

1. What is the binomial theorem and expansion?

The binomial theorem and expansion is a mathematical concept that allows us to expand a binomial expression raised to any power. It is typically written as (a + b)^n, where a and b are constants and n is the power.

2. How is the binomial theorem and expansion used in real-world applications?

The binomial theorem and expansion is used in fields such as statistics, physics, and engineering to solve problems involving probability, geometric series, and power series. It is also used in finance to calculate compound interest.

3. Can you provide an example of using the binomial theorem and expansion?

Sure, let's say we have the expression (2x + 3)^4. To expand this using the binomial theorem, we would use the formula (a + b)^n = Σ(nCr)a^(n-r)b^r, where n is the power, r is the term number, and nCr is the combination formula. Plugging in the values, we get (2x)^4 + 4(2x)^3(3)^1 + 6(2x)^2(3)^2 + 4(2x)^1(3)^3 + (3)^4. Simplifying this, we get 16x^4 + 96x^3 + 216x^2 + 216x + 81.

4. Are there any limitations or restrictions when using the binomial theorem and expansion?

Yes, the binomial theorem and expansion can only be used when the power n is a positive integer and the binomial expression is in the form (a + b)^n. It also requires knowledge of the combination formula and the ability to simplify complex expressions.

5. How does the binomial theorem and expansion relate to Pascal's triangle?

Pascal's triangle is a triangular arrangement of numbers that can be used to determine the coefficients in the expansion of a binomial expression. The coefficients are found in the nth row of the triangle, with the first term being the coefficient of a^n and the last term being the coefficient of b^n. This makes it easier to expand binomial expressions with larger powers.

Similar threads

Replies
1
Views
575
  • General Math
Replies
8
Views
1K
Replies
3
Views
530
  • General Math
Replies
1
Views
2K
  • Precalculus Mathematics Homework Help
Replies
3
Views
107
Replies
12
Views
813
  • General Math
Replies
7
Views
1K
  • General Math
Replies
2
Views
1K
  • General Math
Replies
2
Views
984
Back
Top