Binomial Theorem. 133 



Therefore, we have proved that if the laWvS of exponents and 

 coefficients hold in the product of 71 factors, they will hold also in 

 the product of ;z+ i factors. 



But they have been proved by actual multiplication to hold 

 when four factors are multiplied together, therefore they hold 

 when five factors are multiplied together, and if they hold when 

 five factors are multiplied together they must hold when six are 

 multiplied together, ana so on indefinitely. Hence the laws hold 

 universally » 



4. Deduction of the Binomial Formula. 

 We have now proved that the equation 



(x-\-a^)(x-\-a^J(x-\-ar,J . . . (x-j-a„._^)(x-^a„) 

 = .r" + (^<2i +(^2 +^3+ . . . -ra„)x"~'^ 



-\-(a^a2-\-ci.'^a-^-^a^a^-\- . . . ~\-a„_^a„)x"~'^ 



-T(a^a2Ci.^-\-a^a2Ci.'^-\-ciiCi2ar^-{- . . -\-a„_^a„_-^a„)x"~^ 

 + . . . -{-a^a^a,^a^ . . a„ 

 is true for all positive values of n. 



Since a^^, a^, a^^, a^, . . . a„ are any numbers whatever, we 

 may assume that they are all alike and we may suppose each 

 equal to the quantity a. Then each of the factors in the left- 

 hand side of the above equation will become equal to x-\-a, and 

 consequently the left-hand member will become 



(x-\-a)". 

 On the right-hand side of the equation the term x" remains un- 

 changed. I^n the second term the parenthesis becomes the sum 

 of n «'s ; that is, it is equal to iia, so that the second term itself 

 becomes 



7iax"-^. 

 In the third term the parenthesis reduces to the sum of as manj^ 

 <2^'s as there are groups of 71 things taken two at a time : that is, 



7l( 71 — - 1 ) 



the parenthesis becomes a", so that the third term itself 



1.2 



becomes 



1.2 . 



In the fourth term the parenthesis reduces to the sum of as 

 many a^'s as there are groups of 71 things taken three at a time ; 



