CHAPTER VIII. 



PROGRESSIONS. 



1. Definitions. An Arithmetical Progression is a series of 

 terms such that each differs from the preceding by a fixed quan- 

 tity, called the common difference. The following are examples : 



7 + 9+11 + 13+15+ . • • 

 31 + 26 + 2i-fi6-f 11+ . . . 



(x—j')-\-x-{-(x-{-j)-{- . . . 



The first ana last terms of any given progression are called the 

 Extremes, and the other terms the Means. 



2. To Find the /zth Term of an Arithmeticai, Pro- 

 ORESSiON. Represent the first term of the progression hy a and 

 the common diflfereiice by d. Then we have 

 Nui7iberofterm. i. 2. 3. 4. 5. 

 Progression!. a -\- (a-\-d) -\- (a-\-2d) + (a + ^f*^) + (a-\-^d), etc. 



We notice that by the nature of the progression every time the 

 number of terms is increased by i the coefficient of d is increased 

 \>y I also ; hence to get the 7i th term from the 5th term, the com- 

 mon difference must be added to it n — ^ times. Whence, 

 representing the ;^th term by /, l=a-\-^d-\-(7i — ^)d, or 



l=a-\-(7i-i)d. (1) 



3. To FIND THE Sum of n Terms of an Arithmeticai. 

 Progression. Representing the sum of the arithmetical pro- 

 gression by s, we have 



s=a-^(a + d)^(a + 2d)^(a + ^d)+ ...+/, (i) 



or, writing this progression in reverse order, we have 



s=l-^(I-d) + (l-2d) + (l-^d)+ . . . ^a (2) 



Now adding (i) and (2) together term for term, noticing that 

 the common difference vanishes, we have 



2s=(a-^l) + (a + l)-^(a^l)^(a + l)+ . . . +(a + l). 



