Binomial Theorem Summary

• A sentence is called a statement if it can be adjudged as true or false. Every statement is a sentence, but a sentence may or may not be a statement.

• A statement involving natural number n is generally denoted by P(n).

• Principle of mathematical induction states that if P(n) is a statement involving natural number n and

- P(1) is true, i.e., the statement is true for n=1.

          - Truth of P(k) implies the truth of P(k+1) i.e., the statement is true for n = k+1 assuming it to be true for n = k, then the statement P(n) is true for all natural numbers.

• A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'.

• The binomial theorem for natural numbers states that

Here, a and b may be any numbers.

• Pascal's triangle

The coefficients of various terms in (a+b)ⁿ for different values of n follows the pattern given below:

                  

• General term

For 0 < r < n, T_r+1 in the expression of (a + b)ⁿ is given by T_r+1 = ⁿC_ra^n-rb^r.

• (r + 1)^th term at the end in the expansion of (a+b)ⁿ is same as the (r + 1)^th term at the beginning in (b+a)ⁿ.

• Middle terms

- If n is an even natural number, then there is only one middle term in the expansion of (a + b)ⁿ and is given by

- If n is an odd natural number, then there are two middle terms in the expansion of (a + b)n and are given by

• i) The sum of all binomial coefficients in the expansion of (1+x)ⁿ is 2ⁿ.

ii) The sum of all even binomial coefficients in the expression of (1+x)ⁿ is 2^n-1.

iii) The sum of all odd binomial coefficients in the expansion of

The last terms in (ii) and (iii) depends upon the fact whether n is even or odd.

• The binomial theorem for fractional index states that

• General term

For r 0, T_r+1 in the expansion of (1+x)ⁿ, |x|<1,n Q is given by

• If x be so small that its squares and higher powers may be neglected, then (1+x)ⁿ= 1 + nx (approximately).

• If x be so small that its cube and higher powers may be neglected, then