no Algebra. 



gression is 15 and the sum of their squares is 83 ; find the com- 

 mon difference. 



22. There are two arithmetrical series which have the same 

 common difference ; the first terms are 3 and 5 respectively and 

 the sum of seven terms of the one is to the sum of seven terms of 

 the other as 2 to 3. Determine the series. 



7. Definitions. A Geo77tetrical Progression is 3. s&ri&s oi t^rms 

 such that each is the product of the preceding by a fixed factor 

 called the Ratio. The following are examples : 



3 + 6+12 + 24 + 48, etc. 

 100+50 + 25+ 125- + 6I-, etc. 

 i + i+i + xV+sV," etc. 

 i + i + TVH-2V+4V etc._ 

 The first and last terms of any progression are often called the 

 Extremes and the remaining terms the Means. 



8. To Find the 7i th Term. I^et a represent the first term 

 of the geometrical progression and r the ratio. Then the pro- 

 gression may be written : 



Number of term. i. 2. 3. 4. 5. 

 Progression. a + ar-\- ar^-\- ar^ + ar^. 



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

 the number of terms is increased i the exponent of r is increased 

 by I also ; hence to get the ?i th term from the 5th term it must 

 be multiplied by the ratio n — ^ times. Whence, reoresenting the 

 nth term by /, and /=(ar^)(n—T,), 

 or 



/=ar"-\ (i) 



9. To Find the Sum of 71 Terms. Represnting the sum of 

 the geometrical progression by ^ we have 



s=a-\-ar-\-ar^-\-ar^-\- . . . -\-ar"~^-\-ar"~\ (i) 

 Multiplying this equation through by r— 1 , we obtain 



(r — i)s==a7-" — a. 

 Whence 



ar" — a 



(2) 



r—\ 



Now ar" = r(ar"~^). Therefore, since ar"~^=l, 



ar"=al. 



