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.

• 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 the value 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 results which are applied in binomial theorem problems are 

1.  ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

2.  ⁿC_r = n/r (^n-1C_r-1)

3.  ⁿC_r/ⁿC_r-1 = (n – r + 1)/r

4. [(n+1)/(r+1)] ⁿC_r = ⁿ⁺¹C_r+1

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

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

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

• An important result:

For sums involving  product of two binomial coefficients the folliwng result is used:

ⁿC₀ ^mC_k + ⁿC₁^mC_k-1 + ⁿC₂^mC_k-2 + ….  + ⁿC_k^mC₀ = ^{m + n}C_k

• In order to find the integral part of the expression N = (a + √b)ⁿ (n ∈ N)

Step 1: Consider N’ = (a -√b)ⁿ or (√b – a)ⁿ according as a > √b or √b > a.

Step 2: Use N + N’ or N – N’ such that the result is an integer.

Step 3: Use the fact that N = I + t, where I stands for [N] and t for {N}.

• When n is a negative integer or a fraction then the expansion of a binomial is possible only when:

(a) its first termis 1.

(b) the second term is numerically less than 1.

• Hence, when n does not belong to n and |x| < 1, then it states that

(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + …. n(n-1).. (n – r + 1)x^r/r! + … ∞

• General term:

T_r+1 = n(n-1)(n-2) .. .(n – r + 1). x^r/r!

In this expansion there are infinite terms.

In this case, the differnet terms cannot be expressed as ⁿC₀, ⁿC₁, … because n is not a positive integer.

• In order to find the sum of an infinite series, we can compare the given infinite series with the expansion of (1+x)ⁿ = 1 + nx + n(n-1)/2! x² + … and by finding the value of x and n and putting in (1+x)ⁿ the sum of series is determined.