Mathematics

Binomial Theorem Session 1

Session Objectives

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

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

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

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

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

Question

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

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

Question

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

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

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 = ?

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)

Question

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

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

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

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

Question

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

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

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

Question

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

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

Class Test

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

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

Class Exercise 3 Solution : Now as

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 =

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

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

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.

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

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

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

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

Solution Cont.

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

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 :

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

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

Solution Cont. Similarly ...(ii)

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

Thank you