136 Algkbra. 



Multiplying numerator and denominator of the coefficient by 

 1 n—/^-\- 1 , this becomes 



r— I \7i — r-\- I 



10. Theorem. /?z the expansion of (x^a)" the coefficie?it of 

 the rth term from the beginning; equals the coefficient of the rth tei7n 

 from the e?id. 



Since there are n-\-i terms all together (Art. 8), the /th term 

 from the end has 7i-\-i—t, or 7i — t-\-i, terms before it. Hence 

 the /th term from the end is the same as the ;^ — /4-2 th term from 

 the beginning. From the preceding article the ;z — /+2th term 

 equals 



\n 



\n—t-\-i \t—i 



But from the preceding article the/th term from the beginning 

 equals 



|/— I I ;•? — /'+ I 

 It is plainly seen that the coefficients are identical. 



11. Expansion of (x—a.y 



If we substitute —a for a in the binomial formula^we will 



obtain the following result for the expansion o^ x—a : 



/ w, „ » T ■ n(n—\) , 71(71— I )( 71— 2) 



(x—a)"=x"—nax"-'-\-- ^ a^x"'^ * ^- ^ a'x"'-^^- . . 



1.2 1.2.3 



12. Theorem, hi the bi7i07nial for7nula the S7i7n of the coeffi- 

 cients of the even ter7ns equals the sum of the coefficie7its of the oda 

 terms. 



In the expansion of (x—a)"'pnt x=i and a=i. We then 

 obtain 



, 7i(n—\) n(n — i)(n—2) , 



0= I — ;/ + -^ ^ ^-^ ^ + etc. , 



1.2 1.2.3 



which shows, since the negative on the right side of this equation 



must equal the positive, that the sum of the coefficients of the 



first, third, fifth, . . . terms equals the sum of the coefficients of 



the second, fourth, sixth, . . . terms. 



