In any arithmetical progression, the common difference is equal to the last term, minus the first term divided by the number of terms less one. If the last term is less than the first, the common difference will be negative, as it should be. EXAMPLES. 1. The first term of a progression is 4 the last term 16, and the number of terms considered 5: what is the common 2. The first term of a progression is 22, the last term 4, and the number of terms considered 10: what is the common difference? Ans. 2. 179. By the aid of the last principle deduced, we the following problem, viz.: can solve To find a number m of arithmetical means between two given numbers a and b. To solve this problem, it is first necessary to find the common difference. Now, we may regard a as the first term of an arithmetical progression, b as the last term, and the required means as intermediate terms. The number of terms considered, of this progression, will be expressed by m +2. Now, by substituting in the above formula, b for 1, and m + 2 for n, it becomes that is, the common difference of the required progression is obtained by dividing the difference between the last and first terms by one more than the required number of means. Having obtained the common difference, form the second term of the progression, or the first arithmetical mean, by adding d, or 1. Find 3 arithmetical means between 2 and 18. The formula 2. Find 12 arithmetical means between 77 and 12. The formula 180. If the same number of arithmetical means be inserted between the terms of a progression, taken two and two, these terms, and the arithmetical means together, will form one and the same progression. For, let a.b.c.e. f.... be the proposed progression, and m the number of means to be inserted between a and b, b and c, c and e. ... From what has just been said, the common difference of each partial progression will be expressed by which are equal to each other, since, a, b, c, are in pro ... gression: therefore, the common difference is the same in each of the partial progressions; and since the last term of the first, forms the first term of the second, &c., we may conclude that all of these partial progressions form a single progression. GENERAL EXAMPLES. 1. Find the sum of the first fifty terms of the progression 2.9.16.23 For the 50th term, we have 2. Find the 100th term of the series 2.9. 16. 23 . . Ans. 695. 3. Find the sum of 100 terms of the series 1.3.5 7.9... Ans. 10000. 4. The greatest term considered is 70, the common difference 3, and the number of terms 21: what is the least term and the sum of the terms? Ans. Least term 10; sum of terms 840. 5. The first term of a decreasing arithmetical progression is 10, the common difference is , and the number of terms required the sum of the terms. 21 Ans. 140. 6. In a progression by differences, having given the common difference 6, the last term 185, and the sum of the terms 2945: find the first term, and the number of terms. Ans. First term 5; number of terms 31. 7. Find 9 arithmetical means between each antecedent and consequent of the progression 2.5.8.11.14 Ans. d= 0.3. 8. Find the number of men contained in a triangular battalion, the first rank containing 1 man, the second 2, the third 3, and so on to the nth, which contains n. In other words, find the expression for the sum of the natural numbers 1, 2, 3, . . . from 1 to n, inclusively. Ans. S= n(n + 1) 2 9. Find the sum of the first n terms of the progression of uneven numbers 1, 3, 5, 7, 9 ... Ans. Sn3. 10. One hundred stones being placed on the ground, in a straight line, at the distance of two yards from each other, how far will a person travel who shall bring them one by one to a basket, placed at two yards from the first stone? Ans. 11 miles 840 yards. Of Ratio and Geometrical Proportion. 181. The RATIO of one quantity to another, is the quotient which arises from dividing the second by the first. Thus, the 182. Two quantities are said to be proportional, or in proportion, when their ratio remains the same, while the quantities themselves undergo changes of value. Thus, if the ratio of a to b remains the same, while a and b undergo changes of value, then a is said to be proportional to b. 183. Four quantities are in proportion, when the ratio of the first to the second, is equal to the ratio of the third to the fourth. Thus, if α the quantities a, b, c and d, are said to be in proportion. We generally express that these quantities are proportional by writing them as follows: a: b : : c: d. This algebraic expression is read, a is to b, as c is to d, and is called a proportion. 184. The quantities compared, are called terms of the pro portion. The first and fourth terms are called the extremes, the second and third are called the means; the first and third are called ontecedents, the second and fourth are called consequents, and the fourth is said to be a fourth proportional to the other three. If the second and third terms are the same, either of these is said to be a mean proportional between the other two. Thus, in the proportion b is a mean proportional between a and c, and c is said to be a third proportional to a and b. 185. Two quantities are reciprocally proportional when one is proportional to the reciprocal of the other. Geometrical Progression. 186. A GEOMETRICAL PROGRESSION is a series of terms, each of which is derived from the preceding one, by multiplying it by a constant quantity, called the ratio of the progression. If the ratio is greater than 1, each term is greater than the preceding one, and the progression is said to be increasing. If the ratio is less than 1, each term is less than the pre. ceding one, and the progression is said to be decreasing. Thus, 3, 6, 12, 24, . . . &c., is an increasing progression. It may be observed that a geometrical progression is a continued proportion in which each term is a mean proportional between the preceding and succeeding terms. 187. Let r designate the ratio of a geometrical progression, a b c d, . . &c. We deduce from the definition of a progression the follow ing equations: |