Monday, 24 July 2023

Binomial Expansion Formulas

 

Binomial Expansion Formulas

To find binomial coefficients, we can also use Pascal’s Triangle.

Pascal's triangle for Binomial Expansion

Binomial Expansion

Important points to remember

  • The total number of terms in the expansion of (x+y)n  is (n+1)
  • The sum of exponents of x and y is always n.
  • nC0, nC1, nC2, … .., nCn are called binomial coefficients and also represented by C0, C1, C2, ….., Cn
  • The binomial coefficients, which are equidistant from the beginning and from the ending, are equal, i.e., nC= nCn, nC= nCn-1 , nC= nCn-2 ,….. etc.

Binomial Theorem for Any Index

Let n be a rational number and x be a real number, such that | x | < 1. Then,

Binomial Theorem for any Index If the rational number and -1 < x <1, then,

  • (1 − x)-1 = 1 + x + x+ x+ … + x+ … ∞
  • (1 + x)-1 = 1 – x + x– x+ … (−1)xr+ … ∞
  • (1 − x)-2 = 1 + 2x + 3x2 − 4x3 + … + (r + 1)xr + … ∞
  •  (1 + x)-2 = 1 − 2x + 3x− 4x+ … + (−1)(r + 1)x+ … ∞






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