Revision Notes: Binomial Theorem & its Simple Applications | Mathematics (Maths) for JEE Main & Advanced PDF Download

Revision Notes On Binomial Theorem

• If x, y ∈ R and n ∈ N, then the binomial theorem states that  (x+y)ⁿ = ⁿC₀xⁿ + ⁿC₁ x^n-1y+ ⁿC₂ x^n-2 y² +…… … .. + ⁿC_rx^n-r y^r + ….. + ⁿC_nyⁿ which can be written as ΣⁿC_rx^n-ry^r. This is also called as the binomial theorem formula which is used for solving many problems.

• Some chief properties of binomial expansion of the term (x+y)ⁿ:
  • The number of terms in the expansion is (n+1) i.e. it is one more than the index.
  • The sum of indices of x and y is always n.
  • The binomial coefficients of the terms which are equidistant from the starting and the end are always equal. The simple reason behind this is C(n, r) = C(n, n-r) which gives C(n, n) C(n, 1) = C(n, n-1) C(n, 2) = C(n, n-2).
• Such an expansion always follows a simple rule which is:
  • The subscript of C i.e. the lower suffix of C is always equal to the index of y.
  • Index of x = n – (lower suffix of C).

• The (r +1)^thterm in the expansion of expression (x+y)ⁿ is called the general term and is given by T_r+1 = ⁿC_rx^n-ry^r

• The term independent of x is obviously without x and is that value of r for which the exponent of x is zero.

• The middle term of the binomial coefficient depends on thevalue of n. There can be two different cases according to whether n is even or n is odd.

  • If n is even, then the total number of terms are odd and in that case there is a single middle term which is (n/2 +1)^th and is given by ⁿC_n/2 a^n/2 x^n/2.
  • On the other hand, if n is odd, the total number of terms is even and then there are two middle terms [(n+1)/2]^thand [(n+3)/2]^th which are equal to ⁿC_(n-1)/2 a^(n+1)/2 x^(n-1)/2 and ⁿC_(n+1)/2 a^(n-1)/2 x^(n+1)/2

• The binomial coefficient of the middle term is the greatest binomial coefficient of the expansion.

• Some of the standard binomial theorem formulas which should be memorized are listed below:

  • C₀ + C₁ + C₂ + ….. + C_n= 2ⁿ
  • C₀ + C₂ + C₄ + ….. = C₁ + C₃ + C₅ + ……….= 2^n-1
  • C₀² + C₁² + C₂² + ….. + C_n² = ²ⁿC_n = (2n!)/ n!n!
  • C₀C_r + C₁C_r+1 + C₂C_r+2+ ….. + C_n-rC_n=(2n!)/ (n+r)!(n-r)!
  • Another result that is applied in binomial theorem problems is ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r
  • We can also replace ^mC₀ by ^m+1C₀ because numerical value of both is same i.e. 1. Similarly we can replace ^mC_m by ^m+1C_m+1.
  • Note that (2n!) = 2ⁿ. n! [1.3.5. … (2n-1)]

• In order to compute numerically greatest term in a binomial expansion of (1+x)ⁿ, find T_r+1 / T_r= (n – r + 1)x/r. Then put the absolute value of x and find the value of r which is consistent with the inequality T_r+1 / T_r> 1.

• If the index n is other than a positive integer such as a negative integer or fraction, then the number of terms in the expansion of (1+x)ⁿis infinite.

• The expansions in ascending powers of x are valid only if x is small. If x is large, i.e. |x| > 1 then it is convenient to expand in powers of 1/x which is then small.

• The binomial expansion for the nth degree polynomial is given by: