Mathematics

1. Mathematics

2. Session Binomial Theorem Session 1

3. Session Objectives

4. Session Objective • Binomial theorem for positive integral index • Binomial coefficients — Pascal’s triangle • Special cases • (i) General term • (ii) Middle term • (iii) Greatest coefficient • (iv) Coefficient of xp • (v) Term dependent of x • (vi) Greatest term

5. Binomial Theorem for positive integral index Any expression containing two terms only is called binomialexpression eg. a+b, 1 + ab etc For positive integer n Binomial theorem where are called binomial coefficients. numerator contains r factors

6. 1 1 1 1 1 2 3 3 1 1 1 4 6 4 1 1 5 10 10 5 1 0C0 1 1 C C 0 1 2 C 2 2 1 C C 0 2 3 3 C C 3 1 2 3 C C 0 3 4 4 4 4 C C C C 4 C 4 2 1 3 0 5 5 5 5 5 5 C C C C C C 5 0 1 2 3 4 Pascal’s Triangle

7. Observations from binomial theorem • (a+b)n has n+1 terms as 0  r  n • Sum of indeces of a and b of each term in above expansion is n • Coefficients of terms equidistant from beginning and end is same as ncr = ncn-r

8. Special cases of binomial theorem in ascending powers of x in descending powers of x

9. Question

10. Expand (x + y)4+(x - y)4 and hence find the value of Illustrative Example Solution : Similarly =34

11. General term of (a + b)n n+1 terms kth term from end is (n-k+2)th term from beginning

12. Question

13. Find the 6th term in the expansion of and its 4th term from the end. Illustrative Example Solution :

14. Find the 6th term in the expansion of and its 4th term from the end. Illustrative Example Solution : 4th term from end = 9-4+2 = 7th term from beginning i.e. T7

15. Middle term = ? Middle term CaseI: n is even, i.e. number of terms odd only one middle term CaseII: n is odd, i.e. number of terms even, two middle terms

16. Greatest Coefficient CaseI: n even CaseII: n odd

17. Question

18. Find the middle term(s) in the expansion of and hence find greatest coefficient in the expansion Illustrative Example Solution : Number of terms is 7 + 1 = 8 hence 2 middle terms, (7+1)/2 = 4th and (7+3)/2 = 5th

19. Find the middle term(s) in the expansion of and hence find greatest coefficient in the expansion Illustrative Example Solution : Hence Greatest coefficient is

20. Coefficient of xp in the expansion of (f(x) + g(x))n Algorithm Step1: Write general term Tr+1 Step2: Simplify i.e. separate powers of x from coefficient and constants and equate final power of x to p Step3: Find the value of r

21. Term independent of x in (f(x) + g(x))n Algorithm Step1: Write general term Tr+1 Step2: Simplify i.e. separate powers of x from coefficient and constants and equate final power of x to 0 Step3: Find the value of r

22. Question

23. Find the coefficient of x5 in the expansion of and term independent of x Illustrative Example Solution : For coefficient of x5 , 20 - 5r = 5  r = 3 Coefficient of x5 = -32805

24. Term independent of x Solution Cont. For term independent of x i.e. coefficient of x0 , 20 - 5r = 0  r = 4

25. Step2: Solve for r Step3: Solve for r Greatest term in the expansion Algorithm Step1: Find the general term Tr+1 Step4: Now find the common values of r obtained in step 2 and step3

26. Question

27. Illustrative Example Find numerically the greatest term(s) in the expansion of (1+4x)8, when x = 1/3 Solution :

28. r = 5 i.e. 6th term Solution Cont.

29. Class Test

30. Find the term independent of x in the expansion of For the term to be independent of x Class Exercise 1 Solution : Hence sixth term is independent of x and is given by

31. Find (i) the coefficient of x9 (ii) the term independent of x, in the expansion of Class Exercise 2 Solution : i) For Coefficient of x9 , 18-3r = 9  r = 3 hence coefficient of x9 is -28/9 ii) Term independent of x or coefficient of x0,18 – 3r = 0  r = 6

32. Class Exercise 3 Solution : Now as

33. Sum of the coefficients is i.e. odd number of terms Middle term = Class Exercise 4 If the sum of the coefficients in the expansion of (x+y)n is 4096, then prove that the greatest coefficient in the expansion is 924. What will be its middle term? Solution : greatest coefficient will be of the middle term

34. If then prove that ...(i) ...(ii) Adding (i) and (ii) we get Class Exercise 5 Solution : Replace x by –x in above expansion we get

35. Solution Cont. Put x = 1 in above, we get

36. are in AP Class Exercise 6 Let n be a positive integer. If the coefficients of 2nd, 3rd, 4th terms in the expansion of (x+y)n are in AP, then find the value of n. Solution :

37. Show that Hence show that the integral part of is 197. Class Exercise 7 Solution :

38. where Let i.e. and f = fraction part of I = Integral part of Solution Cont. = 2 (8 + 15.4 + 15.2 + 1) = 198 = RHS

39. Now as is an integer lying between 0 and 2 let Integer part of is 197. Solution Cont.

40. Find the value of greatest term in the expansion of Consider Class Exercise 8 Solution : Let Tr+1 be the greatest term

41. Solution Cont.

42. is the greatest term. Solution Cont. r = 7 is the only integer value lying in this interval

43. If O be the sum of odd terms and E that of even terms in the expansion of (x + b)n prove that i) ii) iii) Class Exercise 9 Solution :

44. O - E = (x-b)n 4 OE = Solution Cont.

45. Let the terms be Class Exercise 10 In the expansion of (1+x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n. Solution :

46. Similarly ...(ii) Solution Cont.

47. Solution Cont. From (i) and (ii) n = 12

48. Thank you