Welcome To A Session Binomial Theorem

Business Mathematics Eleventh Edition BY D.C. Sancheti  V.K. Kapoor Chapter 10 Binomial Theorem PowerPoint Presentation by S. B. Bhattacharjee

© Sharadindu Bikash Bhattacharjee Statement : If n is a positive integer, then Proof: (By the method of induction) By actual multiplication we have © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Thus, the theorem is true for n = 2 and n = 3 We assume that the theorem is true for n = m , Multiplying both sides by (a+x), we have © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee The expansions (1) and (2) are exactly of the same form. Thus, if the theorem is true for n = m, it is true for n = m+1. But we have seen that the theorem is true for n = 3 it is true for the value ( 3+1) i.e. 4 and so on. Hence, the theorem is true for all positive integral values of n. © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee General term in the expansion of (a+x)n The general term in the expansion of (a+x)n is given by___ General term in the expansion of (1+x)n The general term in the expansion of (1+x)n is given by ___ © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Example 1 If the co-efficient of x2 and x3 in the expansion of are equal the value of k. Let x2 occur in Tr+1 i.e. (r+1)th term Since x2 occurs in Tr+1 xr =x2 r = 2 Again, let x3 occur in Tr+1 Tr+1 = 9cr.39-r. (kx)r = 9cr. 39-r.kr.xr © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Since x3 occurs in Tr+1 xr = x3 r = 3 Co-efficient of x3 = 9c3.39-3. k3 = 9c3.36.k3 According to the question, © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Example 2 If the 21st and 22nd terms in the expansion of (1+x)44 are equal, find the value of x. 21st term i.e. T21= T20+1 = 44c20.x20 22ndterm i.e. T22 = T21+1= 44c21.x21. According to the question, © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Example 3 Find the term independent of x in the expansion of Answer: Let (r+1)th term i.e. Tr+1 be the required term. © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Since the term is independent of x, there must be no x. (8+1)th i.e. 9th term is independent of x. © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Example 4 Find the co-efficient of in the expansion of Answer: Let (r+1) term i.e. Tr+1 contain © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Since this term contains Continued… © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Example - 5 a) b) Continued… © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Continued… © Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee

© Sharadindu Bikash Bhattacharjee Thank you For Attending the Session © Sharadindu Bikash Bhattacharjee