1'RO 



PRO 





by the differenced 



But since this progression is only a compound of two series, viz. 



,. C Equals fl, a, a, a, a, & 



01 Arith. proportionals 0, -f- 6, -f- 2 6, -j- 36, -j- 46, > " 



If 1. 3, 5, 7, 9, &c. a, a -f b, a -{- 2 6, 

 -f 3 6, &c. a, a 6, a 2 6, a 3 6, 

 &c, are in arithmetical progression. Hence 

 it is manifest, that if a be the first term, 

 and a -\- b the second, a +2 6 is the third, 

 a + 3 b the fourth, &c. and a -f n 1~ 

 the n rh or last term. 



" The sum of a series of quantities in 

 arithmetical progression is found by mul- 

 tiplying the sum of the first and last 

 terms by half the number of terms." 



Let a be the first term, b the common 

 difference, n the number of terms, and s 

 the sum of the series : Then, 



a -j-a+6 



s, or, 



Sum,2a+ 



--J-&C. to n terms, = 2s, 



or, 2 a-f-71 1.6 X 



n-l.* X |. 



Any three of the quantities s,a,n, 6, 

 being given, the fourth may be found from 



the equation s = 2 a -f- n 1.6 X ?. 



JEar. 1. To find the sum of 18 terms of 

 the series 1, 3, 5, 7, &c. 



Here a = 1, 6 = 2, n = 18 ; there- 

 fore, * = 2~+ 34 x 9 = 324. 



Ex. 2. Required the sum of 9 terms of 

 the series 11, 9, 7, 5, &c. 



In this case a = 11, b = 2, n = 9 ; 

 therefore s = 22 16 X ^ == 6 x| =27. 



lir. If the first term of an arithmetical 

 progression be 14, and the sum of 8 

 terms be 28, what is the common differ- 

 ence ? 



Since 2 c -j- n 1.6 x - = * 

 n I .6 = 



therefore, 6 = 

 VOL V 



" 



proposed, s = 28, a = 14, n =. 8, there- 



the case 



Hence, the series is 14, 11, 8, 5, &c. 

 PROGRESSION geometrical. Quantities 

 are said to be in geometrical progression, 

 or continual proportion, when the first is 

 to the second, as the second to the third, 

 and as the third to the fourth, &c. that is, 

 when every succeeding terns is a certain 

 multiple, or part of the preceding term. 

 If a be the first term, and ar the second, 

 the series will be a, ar, ar 1 , aH, ar4, &c. 

 For a : ar :: ar : rtr 1 :: ar* : ar3, &c. 



The constant multiplier is called the 

 common ratio, and it may be found by 

 dividing the second term by the first. 



" If quantities be in geometrical pro- 

 gression, their differences are in geome- 

 trical progression." 



Let a, ar, ar* r3, aK, &c. be the quan- 

 tities ; their differences, ar a, ar 1 ar, 

 ari or* ar* ar3, &c. form a geo- 

 metrical progression, whose first term is 

 ar a, and common ratio r. 



" Quantities in geometrical progression 

 are proportional to their differences." 



For a : ar :: ar a : or 1 ar :; ar 1 

 ar : ar> ar a , &c. 



*'In any geometrical progression, the 

 first term is to the third, as the square of 

 the first to the square of the second." 



Let a, ar, ar 1 , &c. be the progression ; 

 then a : ar 1 :: a a : a 2 r*. 



Hence it appears, that the duplicate 

 ratio of two quantities (Euc. Def. 10. 5.), 

 is the ratio of their squares. 



In the same manner it may be shown, 

 that the first term is to the rc-f-l th term, 

 as the first raised to the ?i' h power, to the 

 second raised to the same power. 



" If any terms be taken at equal in- 

 tervals in a geometrical progression, they 

 will be in geometrical progression." 



Let a, a r....ar n ...... ar 2 " ....... a r3 ...... &c. be 



the progression, then a, am, a/2, ar 3 ^ 

 &c. are at the interval of n terms, and 

 form a geometrical progression, whose 

 common ratio is rn. 



" If the two extremes, and the number 

 of terms in a geometrical progression be 

 given, the means may be found." 



Let a and b be the extremes, 71 the num- 



