Progressions. 107 



If the number of terms in the original progression be called n^ 



this becomes 



2s=^ti(a-\-l)^ 



whence s=\n(a-\-l). 



4. To Insert any Number of Arithmetical Means be- 

 tween TWO GIVEN Quantities. Suppose we are to insert p 

 arithmetical means between the two terms a and /. The whole 

 number of terms in the progression consists of the r means and 

 the two extremes. Hence the number of tenns in the progression 

 is/ + 2. Therefore, substituting in (i), Art. 2, we obtain 



l=a + (p+2—\)d, 

 - / — a 



and now, since the common difference is known, any number of 

 means can be found by repeated additions. 



5. The two equations 

 l=a-\-(n—i)d ■ ' (1) 

 s^\n(a-^l) (2) 



contain five different quantities. If any two of them are unknown 

 and the values of the re^t are given the values of the two un- 

 known can be determined by a solution of the system. As an 

 example, suppose that a and d are unknown and the rest known. 

 Putting X for a and y for d so that the unknown quantities will 

 appear in their usual form, the system becomes 



l=x-{-(n-i)y (s) 



s=\yi(x-^l) (O 



Finding the value of x in each equation the system becomes 



^x=f-(?i-i)y (5} 



|.= --/, (6) 



Whence, equating right-hand members of (s) and (6), we obtain 



2nl—2S , . 



whence y— -. . (7^ 



n{n—\) 



