GEOMETRICAL PROGRESSION.] 



MATHEMATICS. ALGEBRA. 



479 



We have already seen that the last term I is ar*~ ', so that 

 rl = or", . '. the first expression for S may le written 



and this formula, expressed in words, furnishes the fol- 

 lowing rule to obtain the sum. 



RULE. Multiply the last term by the ratio ; from the 

 product subtract the first term, and divide the remainder 

 by the ratio minus 1. 



This rule applies, of course, whether the ratio bo whole 

 ' or fractional, positive or negative. When it is a. proper 

 \ fraction, the series is a decreasing one ; and if carried on 

 to an unlimited extent, must supply terms approaching 

 nearer and nearer to zero or nothing, the terms at length 

 differing from by quantities too small to be assigned. 

 Although we cannot write down such an infinite number 

 of terms, yet we may affirm, with confidence, that they 

 are all comprehended between these extreme limits, 

 namely, the first term o, and ; with these limits in- 

 cluded ; for including the in the series cannot affect the 

 value of the sum. In finding this sum we may therefore 

 regard as the last term of the infinite decreasing series ; 

 EO tliat, for all such series, the formula above, I being 0, 

 v.-ill bo 



B< 



1 r 



(2); 



and from this you see strange as the thing may appear 

 tliat it is easier to find the entire sum of an infinite 

 geometrical decreasing series, than to find the sum of 

 or four of its leading terms ; because the formula 

 (1) involves more calculation than the formula (2). For 

 distinction sake, it is usual to replace the symbol S, for 

 the sum of a finite series, S when the series is infinite ; 

 that is, to write the formulae (1) and (2) thus : 



1. Required the sixth term and the sum of six terms 

 of the series 1 -f- 2 + 4 + &c., in which o = 1, r = 2, 



and n = 6, I = ar-l = 2* = 32 ; S = ^| = 2 X 32 



1 = 63. 



2. Required the sum of five terms of tho series 1 4 

 = 16 64 + etc., in which r = 4. 



Here I = 256 ; S = ^=f- = 1^ = 205. 



3. Required the sum to seven terms, as also to infinity 

 of each of the series, 1 + J + J + *-i au( l 1 } + } 

 etc. In the first series r = , n = 7 .". I = ar" '= (J) 



= 64' 



S 

 ^ 



I a = f 1_ 1\ _: * ''"- V 2 = 1' 



1 VL23 ' 2 128 Gl" 



This is the sum of seven terms. When tho number of 

 terms is infinite, that is when n = , * the sum is 



1 r 



= 1 



'2; 



so tliat the sum of the infinite series is just double of the 

 first term ; and you see how much more easily the sum 

 of an infinite number of terms ia found than the sum of 

 seven terms. 



In the second series r = -, . . for the seventh term 



Here, when tho means are supplied, there will be seven 



terms . . n = 7 ; also a = 2, and I = - ; but I = ar" ~ l 



82 



= 2rV. 2r= * .... r6 = _ 1 -.-. r = +l. 

 32 61 X 2 



Consequently the five means are + 1 , j, -}- J, i + A. so that 

 the complete seven terms are either 2 -f- 1 + \ i + i + \ + iV 



EXAMPLES FOR EXERCISE. 



1. Required the sum of five terms of tho series 1 + 2 3 

 + 2* +2" + etc. 



2. Eight terms of 1 + } 1 + <fec. 



3. Ten terms of 1 + 2 + 4 -f 8 -f &o. 



4 What is the geometrical mean between 6 and 54 1 



5. Insert two geometrical means between 2 and 54. 



6. Sum the series ^ + jgj + TQ^ + <fec., to affinity ; 



that is, find the true fractional value of the recurring 

 decimal '1111. . . . 



7. The first term of a geometrical series is 3, the 

 common ratio 5, and the last term 375 : find the sum. 



8. The first term of a geometrical series is |, and tho 

 common ratio J : find the sum to infinity. 



9. Insert three geometrical means between J and . 

 10. Required the sum of tho infinite series, 



and thence deduce thcs values of 

 of x = 2, and x 2. 



in the particular cases 



Besides the two classes of progressions just treated of, 

 there is a third kind which may here be briefly adverted 

 to, namely, harmonical progression. This name is given 

 to every series of which the terms are the reciprocals of 

 those of an arithmetical progression. Thus tho re- 

 ciprocals of the terms of tho arithmetical series 1 + 2 + 

 3+ .... n give the harmonical series 1 -f J -j- 



J + . . . . . It has been found that musical 



strings, of equal thickness and tension, when their 

 lengths are as the numbers 1, J, J, <tc., are in harmony 

 when sounded together ; and hence the term Jiarmonical 

 progression. 



There is no general formula for the sum of an har- 

 monical series, but any number of harmonical means 

 may be inserted between two given extremes by help of 

 the corresponding problem in arithmetical progression. 

 Thus : 



Insert four harmonical means between 2 and 4. 



First insert four arithmetical means between J and }, 

 from the formula I = o + (n \)d, where a = J, n = 0, 

 andZ = J: we thus get } = i + 5(Z ; .'. <i = (} i) 

 -=- 5 = fa .-. since = $, "by applying tho constant 

 difference TV^, we have, for tho four arithmetical means, 

 the quantities ^, $j, ,/, ,f ; that is $,, f , A, ^ ; and, 

 taking the reciprocals of these, tho required harmonical 

 means are 



2J, 2i, 2f, 3J. 



And in this way yoti will find that two harmonical 

 means, between 1 and 2, are l-fr and li ; and that four, 

 between 2 and 12, are 2j, 3, 4 and 6. " 



