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Mathematics

Mathematics. Session. Binomial Theorem Session 2. Session Objectives. Session Objective. Properties of Binomial Coefficients Binomial theorem for rational index — General term — Special cases 3. Application of binomial theorem — Divisibility — Computation and approximation.

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Mathematics

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  1. Mathematics

  2. Session 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 = 1 + 4 + 6 + 4 + 1 = 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 As sum of all the coefficients is 2n

  7. Properties of Binomial Coefficients 3. Proof: For example:

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

  9. Solution Cont.

  10. Class Exercise - 2 Solution :

  11. Solution Cont. = 0 + 0 + 0 = 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 i) |x| < 1 ii) Number of terms is infinite

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

  14. 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. Class Exercise - 6 Solution :

  15. Validity if General term Solution Cont.

  16. Special Cases

  17. Special Cases

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

  20. is divisible by x – y for all positive integer n Application I Division 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 : = 1000000000000 - 300000000 + 30000 -1 = 999700029999

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

  28. Coefficient of x4 is 1.5 – 2.(–4) + 1.3 = 5 + 8 + 3 = 16 Coefficient of xr is Solution Cont.

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

  30. Solution Cont.

  31. Thank you

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