Methods of Induction and Binomial Theorem



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Contents

Mathematical Induction & Ex. 13.1


Description: Mathematical induction proves a statement for all natural numbers by showing it holds for a base case and a recursive step & Ex.13.1 Solutions.


Ex. 13.1 Part-II


Description: Ex. 13.1 Part-II Solutions


Ex. 13.1 Part-III


Description: Ex. 13.1 Part-III Solutions


Ex. 13.1 Part-IV, Positive Integral Index & Ex.13.2


Description: The binomial theorem expands expressions raised to a positive integral power, showing the coefficients of each term systematically & Ex.13.2 Solutions.


Ex. 13.2 Part-II


Description: Ex. 13.2 Part-II Solutions


General & Middle Term Expansion & Ex.13.3


Description: The general expansion of a binomial theorem computes any term, while the middle term expansion finds the term at the midpoint & Ex.13.3 Solutions.


Ex.13.3 Part -II


Description: Ex.13.3 Part -II Solutions


Ex.13.3 Part -III


Description: Ex.13.3 Part -III Solutions


Negative Index or Fraction & Ex. 13.4


Description: The Binomial Theorem for negative or fractional indices expands expressions into an infinite series, useful for approximations, and applies when the absolute value of x is less than one & Ex. 13.4 Solutions


Binomial Coefficients & Ex. 13.5


Description: Binomial coefficients represent the number of ways to choose k items from n, crucial in combinatorics, probability, and polynomial expansions & Ex. 13.5 Solutions.


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