GEOMETRICAL fROGRESSION. I7J 



GEOMETRICAL PROGRESSION. 



Ant series of numbers, the terms of which gradually 

 increas(^ or decrease by a constant multiplication or divis- 

 ion, is said to be in Gecmetrlcal Progression, Thus, 4, 8, 

 16, 32, 64, &c. and 81, 27, 9, 3, I, &c. are series in geo- 

 metrical progression, the one increasing by a constant mul- 

 tiplication by 2, and the other decreasing by a constant di- 

 vision by 3. 



The number, by which the series is constantly increased 

 or diminished, is called the ratio, 



PROBLEM I. 



Given the first terin, the last term, and the ratio, to find the 

 sum of the series. 



RULE.* 



Multiply the last, term by the ratio, ami from the prod- 

 uct subtract the first term, and the remainder, divided by 

 the ratio less i, will give the sum of the scries. 



EXAMPLES. 



* Demonstration. Take any series whatever, as i, 3, 9, 

 27, 8i, 243, &c. multiply this by the ratio, and it will produce 

 the series 3, 9, 27, 81, 243, 729, &c. Now, let the sum of the 

 proposed series be what it v/ill, it is plain, that the sum of the sec- 

 ond series will be as many times the former sum, as is expressed by 

 the ratio ; subtract the first series from the second, and it will give 

 729 — I ; which is evidently as many times the sum of the first 



scries, as is expressed by the ratio less i ; consequently i_^ =2 



3—1 



sum of the proposed series, and is the rule ; or 729 is the last 



term multiplied by the ratio, i is the first term, and 3 — i is the 



ratio less one ; and the same will hold let the series be what It 



will Q^E. D. 



Note- 



