1. Mathematics

2. Binomial Theorem Session 2

3. Session Objectives

4. Session Objective
Properties of Binomial Coefficients
Binomial theorem for rational index
— General term
— Special cases
3. Application of binomial theorem
— Divisibility
— Computation and approximation

5. Properties of Binomial Coefficients
The sum of the binomial coefficients in the expansion of (1+x)n is 2n i.e.
where Cr = nCr
Proof:
Put x = 1 both sides we get
For example 4C0 + 4C1 + 4C2 + 4C3 + 4C4
= = 16 = 24

6. Properties of Binomial Coefficients
2. Sum of coefficients of the odd terms = Sum of the coefficients of the even terms in (1+x)n = 2n-1 i.e.
Proof:
Put x = –1 in above, we get
Ask students to find 4c0+4c2+4c4 and 4c1+4c3 and show that it is 2^3 = 8
As sum of all the coefficients is 2n

7. Properties of Binomial Coefficients3.
Proof:
For example:

8. Class Exercise - 1 Solution :
If C0, C1, C2,... denote the binomial coefficients in the expansion of (1+x)n then prove that
Solution :

9. Solution Cont.

10. Class Exercise - 2
Solution :

11. Solution Cont.
= = 0

12. Binomial theorem for Rational Index
Let n be a rational number and x be real number such that |x| < 1, then
Conditions of validity: if n is not a whole number
Note that if n is a whole number then given expansion is same as that of binomial theorem for positive integral index
i) |x| < 1 ii) Number of terms is infinite

13. General Term in (1+x)n, n  Q
Expansion of (a + x)n for rational n
Case1:
Case2:

14. Class Exercise - 6 Write the first four terms in the expansion of
For what values of x is this expansion is valid? Also, find the general term in this expansion.
Solution :

15. Solution Cont.
Validity if
General term

16. Special Cases

17. Special Cases
Note here that all these expansions are valid only if |x| < 1.

18. Application I Division = Multiple of x = M(x)
Conclusion: The number by which division is to be made can be x or x2 or x3, but the number in the base is always expressed in the form of 1 + kx.

19. Class Exercise - 9 Solution :
Which of the following expression is divisible by 1225?
(a) (b)
(c) (d)
Solution :
= 1225 k
is divisible by 1225.

20. Application I Division
is divisible by x – y for all positive integer n
Proof:
As each term is divisible by x – y,
xn - yn is divisible by x – y

21. Application II Computation and Approximation Find 99993 exactly
Solution : = =

22. Class Test

23. Class Exercise - 3
Solution :
LHS =
= RHS

24. Class Exercise - 4
Solution :

25. Class Exercise - 4
Solution :
Compare the coefficient xn of both sides

26. Class Exercise - 5
Solution :
LHS =
= RHS

27. Class Exercise - 7 Find the coefficient of x4 in the expansion
of Also find the coefficient of xr
and find its expansion.
Solution :

28. Solution Cont. Coefficient of x4 is
1.5 – 2.(–4) = = 16
Coefficient of xr is

29. Class Exercise - 8
When x is so small that its square and higher powers may be neglected, find
the value of
Solution :
As terms involving x2, x3, neglected

30. Solution Cont.

31. Thank you