GEOMETRICAL PROGRESSION. 1 77 



2. The extremes of a geometrical progression are i and 

 65536, and the ratio 4 5 what is the sum of the series ? 



Ans. 87381. 



3. The extremes of a geometrical series are 1024 and 

 59049, and the ratio is i~ ; what is the sum of the series ? 



Ans. 175099. , 



PROBLEM II. 



Given the first term and the rot'io^ to find any other term 

 assigned, 



RULE.* 



1. Write down a few of the leading terms of the series, 

 and place their indices over them, beginning with a cypher. 



2. Add together the most convenient indices, to make an 

 index less by i than the number expressing the place of 

 the term sought. 



3. Multiply the terms of the geometrical series togeth- 

 er, belonging to those indices, and make the product a 

 dividend. 



4. Raise 



* Demonstration. In example i, where the first term is 

 equal to the ratio, the reason of the rule is evident ; for as every 

 term is some power of the ratio, and the indices point out tjie 

 number of factors, it is plain from the nature of multiplication, 

 that the product of any two terms will be another term corres- 

 ponding with the index, which is the sum of tihe indices standbg 

 over those respective terms. 



And in the second example, where the series does not begin 

 with the ratio, it appears, that every term after the two first con- 

 tams some power ^f the ratio, multiplied into the fiist term, and 

 therefore the rule, in this case, is equally evident. 



Thr 



T 



