Class XI Chapter 8 – Binomial Theorem Maths ______________________________________________________________________________ Assuming that m a occurs in the mn 1 1 m n r r mn r r r r T C a C a Comparing the indices of a in We obtain r = m Therefore, the coefficient of m a m a is r 1 th in T , r 1 m n ! m n ! Cm ....... 1 m! m n m ! m! n! m n Assuming that n a occurs in the mn T C a C a mn 1 1 m n k k k k k k Comparing the indices of a in We obtain k n Therefore, the coefficient of n a term of the expansion m n 1 a k 1 th term of the expansion n a and in m n ! m n ! Cn ....... 2 n! m n n ! n! m! m n is Tk 1 Thus, from (1) and (2), it can be observed that the coefficients of of m n 1 a are equal. , m n 1 a , m a and , we obtain we obtain n a in the expansion Question 10: The coefficients of the r ratio 1:3:5. Find n and r . Solution 10: It is known that T C a b k1 n nk k k Therefore, . k 1 th th 1 , r th Tk 1 and r 1 th terms in the expansion of term, , in the binomial expansion of n r 1 th term in the expansion of x 1 n is n n r 2 r 2 n n r 2 r 1 r 2 1 r2 T C x C x r 1 term in the expansion of n nr r n nr r1 r 1 r T C x C x th r term in the expansion of x 1 n x 1 n is is a b x 1 n is given by are in the Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 8 – Binomial Theorem Maths ______________________________________________________________________________ n r 1 r n 1 1 n n r 1 r r1 r1 T C x C x Therefore, the coefficients of the n r th 1 , r th and r 1 th terms in the expansion of x 1 n n Cr2, Cr 1, and n C r are respectively. Since these coefficients are in the ratio 1:3:5, we obtain n n n n C C r2 r1 r1 1 3 and C n r 1 3 n Cr 5 C 1 ! 1 ! 1 2 ! 1 ! r 2 n! r n r r r n r C r 2 ! n r 2 ! n! r 2 ! n r 2 ! n r 1 ! r 1 n r 2 1 1 r nr2 3 3r 3 n r 2 n 4r 5 0 ...... 1 C ! ! 1 ! ! r 1 n! r n r r r n r n C r 1 ! n r 1 n! r 1 ! n r 1 n r ! n r r n r 1 3 r nr1 5 5r 3n 3r 3 3n 8r 3 0 ....... 2 Multiplying (1) by 3 and subtracting it from (2), we obtain 4r 12 0 r 3 Putting the value of r 12 5 0 n 7 Thus, n 7 and r 3 in (1), we obtain n Question **11**: Prove that the coefficient of the expansion of 2 n 1 x 1 . n x in the expansion of 2 n 1 x is twice the coefficient of n x in Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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Class XI Chapter 12 - Introduction

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