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:

1. The number of terms in the expansion is (n+1) i.e. it is one more than the index.

2. The sum of indices of x and y is always n.

3. 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:

1. The subscript of C i.e. the lower suffix of C is always equal to the index of y.

2. 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.

1. 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.

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:

1. C₀ + C₁ + C₂ + ….. + C_n= 2ⁿ

2. C₀ + C₂ + C₄ + ….. = C₁ + C₃ + C₅ + ……….= 2^n-1

3. 3.    C₀² + C₁² + C₂² + ….. + C_n² = ²ⁿC_n = (2n!)/ n!n!

4. 4.    C₀C_r + C₁C_r+1 + C₂C_r+2+ ….. + C_n-rC_n=(2n!)/ (n+r)!(n-r)!

5. Another result that is applied in binomial theorem problems is ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

6. 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.

7. 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:

         

• Following expansions should be remembered for |x| < 1:

1. (1+x)^-1 = 1 – x + x² – x³ + x⁴ - ….. ∞

2. (1-x)^-1 = 1 + x + x²+ x³ + x⁴+ ….. ∞

3. (1+x)^-2 = 1 – 2x + 3x² – 4x³ + 5x⁴ - ….. ∞

(1-x)^-2 = 1 +2x + 3x²+4x³ + 5x⁴+ ….. ∞