io8 



AlvGKBRA. 



Therefore, restoring a and d in (6) and (-]), 



71 

 J 2nl—2S 



n{n — i) 

 where a and d are expressed in terms of the three known quan- 

 tities, n, /, snd s. In Hke manner we may suppose a7iy two of the 

 five quantities unknown and find their values in terms of the 

 known ones. In all, ten different cases may arise, which are 

 given in the following table, each of which is to be worked by 

 the student. 



No 

 I. 



2. 



3- 

 4. 

 5- 

 6. 



Given. 



a, d, n, 

 /, d, n, 



a, I, 71, 



a, 71, s, 

 71, d, s, 

 I, 71, s. 



7. a, d, I, 



9. 



10. 



a, I, s, 

 a, d, s, 

 I, d, s, 



quired 



I, s, 

 a, s, 



d, s, 

 d, I, 

 a, I, 

 a,d. 



Result. 



l^=a-\-{7i — i)d ; 

 a=l—{ii—i)d\ 

 . I— a 



d= 



71—1 

 2S—2a7l 



7i{n — \) ' 



_s {7i—i)d 

 71 2 ' 



2S 



« = /; 



71 



I — a . 

 71, s, ^2="^ — hi ; 



s=\n[2a-\- {71— i)d\ 

 s^=^^n\_2l— {n — i)d~\. 



^= 



/, 71, / = —^d±:^2ds+{a—fdr:^i-- 



a, 71, a = \d^^{i^\dY—ld^ 



71 {n{7i—\ 



27ll — 2S 



7i{n—\)' 

 {l-a){l-a^d) 



2d 



2S — a —V 



d—2a±:^{2a—dy^Sd8 



2d 



2/+ d±z x/'{2l'-f(f)-'-^^H<U 

 2d 



6. Examples. 



1. Find the sum of q terms of the progsession 3 + 7 -f- 1 1 , etc. 



2. The first term is 96, the common difference —5 ; what is 

 the 1 3 th term ? 



