Binomial Theorem. 135 



second coefficients in the preceding power , third, the sura of the second and 

 third coefficients in the preceding power, and so on. Vieta noticed also the 

 uniformity in the product of binomial factors of the form x-\-a, x-\-h, x-{-c, etc. 

 But Harriot (1560-1621) independently and more fully treated of these prod- 

 ucts in showing the nature of the composition of a rational integral equation. 

 See VI, Art. 1. In this connection it is interesting to note that Han-iot was 

 the first mathematician to transpose all the terms of an equation to the left 

 member. 



The binomial formula as now used ; that is, the expansion of the wth power 

 of a binomial, expressed with factorial coefficientH, was the discovery of Sir 

 Isaac Newton (1642-1727) and for that reason it is commonly called Sir Isaac 

 Newton H Binomial Theorem. 



8. Number of Terms in the Expansion. The exponents 

 o{ a through the binomial formula constitute the following scale : 

 o, I, 2, 3, 4, . . . ?i. 



The number of terms in this scale is 7i-\-i. Therefore the 

 number of terms in the expansion of (x-\-a)" is ii-\-i. 



.9. Value of the tth Term. The value of the rth term in 

 the expansion of (x-\-a)" can be easily found 



By the law of exponents, the exponent of x in the first term is 

 71] in the second, n — i ; in the third, n — 2, and so on; conse- 

 quently in the rth term it is n — (r—\), or n — r-\'i. Also by the 

 law of the exponents, the exponent of a in the second term is i ; 

 in the third term, 2, and so on ; consequently in the rth term it 

 is r— I. So, without the coefficient, the rth term must be 

 a''-'x"-''^\ 



By inspection of the coefficients in the expansion in Art. 4, 

 it is seen that the numerator of the coefficient of any term is the 

 product of the natural numbers from ;/ to a number one greater 

 than the exponent of .r. Since the exponent of x in the rth term 

 has been found to be n — r-\-\, this numerator of the coefficient 

 must be ?^(';^— 1X7^ — 2 j . . . (n — r-\-2). 



An inspection of the binomial fonnula will also show that the 

 denominator of any coefficient is the product of the natural num- 

 bers from unity to a number equal to the exponent of a. Whence 

 the denominator of the coefficient of the rth term must be 

 I, 2, 3, . . . (r—\). Therefore the complete rth term is 

 71(11— \) (71 — 2) . . . (n — r-\-2) ^^,._^ ^ „._,.., . 

 1.2.3.4 . • • T''— !•/> 



