1. Mathematics

2. 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 binomial expression eg. a+b, 1 + ab etc
For positive integer n
Binomial theorem
where
in calculating ncr numerator contains r factors starting with n and decreasing by 1 denominator contains product from 1 to r
Now ask them to calculate 5c2, 7c3 etc.
are called binomial coefficients.
numerator contains r factors

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

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

11. General term of (a + b)n n+1 terms
first term is for r = 0, second for r = 1 and so on
n+1 terms
kth term from end is (n-k+2)th term from beginning

12. Question

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

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

15. 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
Middle term
= ?

16. Greatest Coefficient CaseI: n even CaseII: n odd
Tell the students that in case of even terms(n odd) both the coefficients of middle terms are equal using C(n,r) = C(n,n-r)

17. Question

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

19. Illustrative Example Find the middle term(s) in the expansion
of and hence find greatest
coefficient in the expansion
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. Illustrative Example Find the coefficient of x5 in the expansion
of and term independent of x
Solution :
For coefficient of x5 , r = 5  r = 3
Coefficient of x5 =

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

25. Greatest term in the expansion
Algorithm
Step1: Find the general term Tr+1
Step2: Solve for r
Step3: Solve for r
Tell the students that greatest terms means numerically greatest term
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 :
Here note that r is a integer from 0 to 8

28. Solution Cont. r = 5 i.e. 6th term
Here note that r is a integer from 0 to 8
r = 5 i.e. 6th term

29. Class Test

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

31. Class Exercise 2
Find (i) the coefficient of x9 (ii) the term independent of x, in the expansion of
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. 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 :
Sum of the coefficients is
i.e. odd number of terms
greatest coefficient will be of the middle term
Middle term =

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

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

36. 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 :
are in AP
Ask students why n is not equal to 2 ? Its because if n = 2 total number of terms is 3, and in question we are talking of 2nd,3rd and 4th terms.

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

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

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

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

41. Solution Cont.

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

43. Class Exercise 9
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)
Solution :

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

45. 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 :
Let the terms be

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

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

48. Thank you