What is the binomial expansion of 1 x?

This binomial expansion formula gives the expansion of (1 + x)n where ‘n’ is a rational number. This expansion has an infinite number of terms. (1 + x)n = 1 + n x + [n(n – 1)/2!] x2 + [n(n – 1)(n – 2)/3!]

What is the binomial expansion of x 1 n?

Applying Binomial Theorem, if n is a positive integer, (x-1)^n = x^n – C(n,1) x^(n-1) + C(n,2)x^(n-2) -… .

What are the conditions for binomial expansion?

Properties for the binomial expansion include: the number of terms is one more than n (the exponent ), and the sum of the exponents in each term adds up to n . Applying (nr−1)an−(r−1)br−1 ( n r − 1 ) a n − ( r − 1 ) b r − 1 and (nk)=n! (n−k)!k! ( n k ) = n !

Why was the binomial theorem created?

The binomial theorem provides a simple method for determining the coefficients of each term in the expansion of a binomial with the general equation (A + B)n. Developed by Isaac Newton, this theorem has been used extensively in the areas of probability and statistics .

Why do we use binomial theorem?

The binomial theorem is used heavily in Statistical and Probability Analyses. It is so much useful as our economy depends on Statistical and Probability Analyses. In higher mathematics and calculation, the Binomial Theorem is used in finding roots of equations in higher powers.

Why is binomial series infinite?

From the binomial formula, if we let a = 1 and b = x, we can also obtain the binomial series which is valid for any real number n if |x| < 1. (1 + x)n = 1 + nx + NOTE (1): This is an infinite series, where the binomial theorem deals with a finite expansion.

What is the binomial theorem?

The Binomial Theorem states that, where n is a positive integer: (a + b) n = a n + ( n C 1 )a n-1 b + ( n C 2 )a n-2 b 2 + … + ( n C n-1 )ab n-1 + b n This means use the Binomial theorem to expand the terms in the brackets, but only go as high as x 3.

What are the properties of binomial expansion in math?

Properties of Binomial Expansion. There are a few main properties of binomial expansion. These properties are discussed below: The total number of terms in the expansion of (x+y) n are (n+1) The sum of exponents of x and y is always n. nC 0, nC 1, nC 2, … nC n are called binomial coefficients and also represented by C 0, C 1, C2, … C n.

How do you find the number of terms in a binomial?

The number of terms in the expansion of (x + a) n + (x−a) n are (n+2)/2 if “n” is even or (n+1)/2 if “n” is odd. The number of terms in the expansion of (x + a) n − (x−a) n are (n/2) if “n” is even or (n+1)/2 if “n” is odd. Binomial coefficients refer to the integers which are coefficients in the binomial theorem.

What are the most important properties of binomial coefficients?

Some of the most important properties of binomial coefficients are: C 1 − 2C 2 + 3C 3 − 4C 4 + … + (−1) n-1 C n = 0 for n > 1 Illustration: If (1 + x) 15 = a 0 + a 1 x + . . . . . + a 15 x 15 then, find the value of

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