Binomial Theorem for any index real.pptx

Binomial

1. Binomial Theorem

2. Binomial Expression An algebraic expression containing two terms is called a binomial expression. For example, (a + b), (2x – 3y), etc. are binomial expressions.  

3. BINOMIAL THEOREM FOR POSITIVE INDEX Such formula by which any power of a binomial expression can be expanded in the form of a series is known as Binomial Theorem. For a positive integer n , the expansion is given by ( a+x ) n = n C a n + n C 1 a n–1 x + n C 2 a n-2 x 2 + . . . + n C r a n–r x r + . . . + n C n x n = . where n C , n C 1 , n C 2 , . . . , n C n are called Binomial co- efficients . Similarly (a – x) n = n C a n – n C 1 a n–1 x + n C 2 a n-2 x 2 – . . . + (–1) r n C r a n–r x r + . . . +(–1) n n C n x n i.e. (a – x) n = Replacing a = 1, we get (1 + x) n = n C + n C 1 x+ n C 2 x 2 + . . . + n C r x r + . . . + n C n x n and (1 – x) n = n C – n C 1 x+ n C 2 x 2 – . . . + (–1) r n C r x r + . . . +(–1) n n C n x n   then

4. Observations: There are (n+1) terms in the expansion of (a +x) n . Sum of powers of x and a in each term in the expansion of (a +x) n is constant and equal to n. The general term in the expansion of ( a+x ) n is (r+1) th term given as T r+1 = n C r a n-r x r The p th term from the end = ( n –p + 2) th term from the beginning . Coefficient of x r in expansion of (a + x) n is n C r a n - r x r . n C x = n C y  x = y or x + y = n. In the expansion of (a + x) n and (a –x) n , x r occurs in (r + 1) th term.

5. Illustration 3: If the coefficients of the second, third and fourth terms in the expansion of (1 + x) n are in A.P., show that n = 7. Solution: According to the question n C 1  n C 2  n C 3 are in A.P. n 2 – 9n + 14 = 0  (n – 2)(n – 7) = 0  n = 2 or 7 Since the symbol n C 3 demands that n should be  3 n cannot be 2,  n = 7 only.  

6. MIDDLE TERM There are two cases   When n is even Clearly in this case we have only one middle term namely T n/2 + 1 . Thus middle term in the expansion of (a + x) n will be n C n /2 a n/2 x n/2 term.   (b) When n is odd Clearly in this case we have two middle terms namely . That means the middle terms in the expansion of (a +x) n are and .  

7. Illustration 7: Find the middle term in the expansion of . Solution: There will be two middle terms as n = 9 is an odd number. The middle terms will be and terms. t 5 = 9 C 4 (3x) 5 t 6 = 9 C 5 (3x) 4 .  

8. GREATEST BINOMIAL COEFFICIENT In the binomial expansion of (1 + x) n , when n is even, the greatest binomial coefficient is given by n C n /2 . Similarly if n be odd, the greatest binomial coefficient will be

9. NUMERICALY GREATEST TERM