SECT. XX. OF THE RELATION OF NUMBERS. ANY fet of numbers that conftantly increase or decrease by a common difference are faid to bear a relation to each other, which relation is called progreffion, and is divided into two kinds: arithmetical progreffion and geometrical progreffion. Arithmetical progreffion is, when any fet or feries of numbers conftantly increase or decrease by a given number, called from thence their common difference; such are the - natural orders of numbers, 1, 2, 3, 4, 5, 6, 7, 8, &c. each of which increases by 1; and the fame in inverted order 8, 7, ·6, 5, 4, &c. decrease by 1; 1 is therefore their common -difference. Again, the numbers 2, 5, 8, 11, 14, 17, &c. increase by 3; and 27, 23, 19, 15, 11, 7, &c. decrease by 4: therefore 3 is the common difference of the former, and 4 that of the latter. The numbers themselves that form the series are called the terms of the progreffion. Of every series of arithmetical progreffion having any three of the five following terms the other two may be found readily; 1, the first term or number 3, the number of terms.. 3 called the extremes. PROBLEM PROBLEM I. The first and laft terms, and number of terms, being given, to find the fum of all the terms, and the common difference. Rule. To find the fum of all the terms, multiply the fum of the extremes by the number of terms, and half the produ& will be the answer; and to find the common difference of the terms, divide the difference of the extremes by the number of terms made lefs by 1, and the quotient will be the answer. Example 1. What is the sum of the feries, and the com mon difference of that progreffion, whose first term is 3, laft term 67, and number of terms ३.३ 2)2310 1155 Answer, fum of the feries. In the first operation, 70, the fum of the extremes, is multiplied by 33 the number of terms, and half the product 1155 is the fum of the series, or fum of all the numbers, 31. 52 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, added together. In the second operation the difference of the two extremes 67 and 3, or 64, is divided by 32, which is one lefs than the number of terms, and the quotient 2 is the common differ ence. PROBLEM II. Having the two extremes and the common difference, to find the number of terms. Rule. Divide the difference of the extremes by the come, VOL. I. Ii mon mon difference, and I added to the quotient will be the number of terms. Example 2. What is the number of terms of that progreffion whose extremes are 3 and 67, and common difference 2 ? Thus 64, the difference of the extremes, divided by 2, the common difference, quotes 32, to which adding 1 there is 33, the number of terms; as be feen annexed. may 67 extremes 63} ex 2)64 difference 32 I 33 number of terms. In every arithmetical progreffion the fùm of any two terms is equal to the fum of any two other terms, taken at an equal distance from the former, and on oppofite fides: thus, in the foregoing progreffion, 15 and 19 is equal to 23 and 11; viz. 343 and 31 and 33 is equal to 23 and 41, viz. 64. And the double of any one term is equal to the fum of any two terms taken on each side of it, and at an equal distance from it; thus the double of 25 is equal to 19 and 31, viz. 50. Qu. 3. How many strokes does an English clock strike in 8 days?-Anf. 1248 strokes. Qu. 4. If a traveller go a journey of 10 days, travelling 3 miles the first day, and increafing three miles every day, how many miles will he travel in the 10 days? and how many ' miles the laft day?-Anf. 165 miles in the whole, and 30 miles the last day. Geometrical Progreffion. When any series of numbers increase or decrease by a conftant multiplication or divifion, they are faid to be in geometrical progreffion. Thus the numbers 4, 8, 16, 32, 64, &c. and 243, 81, 27, 9, 3, 1, are in geometrical progreffion, the former increasing by multiplying each preceding number by 2, and the latter decreafing by dividing each preceding number by 3. 5 Thus, -Thus, in geometrical progreffion, the numbers are increafed by multiplication, and decreased by divifion; whereas in arithmetical progreffion they are increased by addition and decreased by fubtraction. The number in geometrical progreffion by which the feries are multiplied or divided, is called the ratio; whereas in arithmetical progreffion the number by which the feries are increased or diminished is called the common difference. In geometrical progreffion, as in arithmetical progreffion, having any three of the following terms, the other two may be readily found. Having the first and last term and the ratio, to find the fum of the terms. Rule. Multiply the laft term by the ratio, and from the product fubtract the first term, and the remainder divided by one less than the ratio will quote the fum of the feries. Example 1. What is the fum of the series of a geometrical progreffion whose extremes are 1 and 65536, and the ratio 4? Here the laft term is multiplied by the ratio 4, the first term 1 fubtracted from the product, and the remainder divided by 3, which is less than the ratio, and the quotient 87381 is the fum of the series. Ii 2 Qu. Qu. 2. What is the fum of a series of numbers, in geometrical progreffion, whofe extremes are 1024 and 59949, and the ratio 3 Anfover 88061. T PROBLEM IL Having the first term and the ratio, to find any other term required. Rule. Write down a few of the leading terms of the feries, and place their indices over them, beginning with a cypher Then add any of those indices together, which will make an index an unit lefs than the number which expreffes the place of the term required. Multiply the terms of the feries together belonging to those indices, and the product will be a dividend, to be divided by the product of the firft term, multiplied by a number an unit less than the number of terms multiplied; and the quotient will be the answer. Example 3. What is the last term of that geometrical feties whose first terin is 3, ratio 2, and number of terms 107 Indices 0, 1, 2, 3, 4, 5, Leading terms 3, 6, 12, 24, 48, 96, 192. *The indices of the terms in this rule are numbers expreffing the place of each term according to the natural order of numbers; thus the index of the firft term is o, that of the fecond term 1, the third term 2, the fourth 3, &c., but when the first term of the feries is equal to the ratio, the indices mult begin with an unit, and the product of the terms will be the answer, without dividing by the product of the firft term, as directed above. In |