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rohan03
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(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.
rohan03 said:Thank you . Can I prove with the help of this :
(1+n)n ≥ 5/2nn- 1/2n n-1
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.
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.
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.
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.
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.