I 



OF QUANTITY AND NUMBER. 7 



Lct_>_, thenji3<_; for.f_= ^ , . > o, and— >_ 



(34); consequently— >--~^. The same demonstration may be ex- 

 b b-\-a 



tended to any number of ratios. 



36. Definition. A series of numbers, formed by the 

 continual addition of the same number to any given num- 

 ber, is called an arithmetical progression. 



2, 5, 8, 11, 14, 17, 20, by adding 3. 

 20, 17, 14, 11, 8, 5, 2, by adding— 3. 



a, a-+-fe, 0+26, a-f-36, a-\-{n — 1).&, in general. 



Scholium. It may be observed that the sum of each pair of the 

 numbers of these equal progressions is 22~2+20zia4-«+ (w — 1).6= 

 2a4-(/i — l).i ; the whole sum 22x 7=(2a-|- (n — 1). b). w, and the sum 



of each, na -f . 6, a being the first term, b the difference, and n 



the number of terms. 



37. Definition. A series of numbers, formed by con- 

 tinual multiplication by a given number, is called a geome- 

 trical progression. 



As 2, 6, 18, 54 ; multiplying 2 continually by 3. 

 a, ab, abb, abbb ; multiplying a by b. 



38. Definition. If one of the terms of a geometri- 

 cal progression is unity, the other terms are called powers 

 of the common multiplier. 



As j'j, ^, \y ^, \, 1, 2, 4, 8, 16, 32. Each term is denoted by placing 

 obliquely over the common multiplier a number expressive of its dis 

 tance from unity, as 8n2^ : negative numbers, implying a contrary 

 situation to positive ones, denote that the term precedes instead of fol- 

 lowing the unit, as ^=2"^. 



By reversing the scries it is obvious that g=:(|y, and 8zi(^)'"^ 



It appears that tlie addition of the indices denoting the places of 



any terms will point out a term which is their product, as 2^x2^zr2*, 



or 8 X 4:z:32 ; and that the subtraction of the index is equivalent to 



division by the term. Hence if a^rzt— 6', «• must be equal to 6^ in 



