extremes and all the intermediate terms. If, then, m repre sents the number of means, m+2 will be the whole number of terms. Substituting m+2 for n in the formula, Art. 322, we have l=a+(m+1)d, or 1-a d= m+1 = -the common difference, whence the required means are easily obtained by addition. 326. The two equations l=a+(n−1)d, s=12 (a+1), contain five quantities, a, l, n, d, s, of which any three being given, the other two can be found. We may therefore have ten different cases, each requiring the determination of two dif ferent formulæ. These formulæ are exhibited in the following table, and should be verified by the student. EXAMPLES. 1. The first term of an arithmetical progression is 2, and the common difference is 4; what is the 10th term? Ans. 38. 2. The first term is 40, and the common difference -3; what is the 10th term? 3. The first term is 1, and the common difference ; what is the 10th term? 4. The first term is 1, and the common difference -; what is the 10th term? 5. The first term is 5, the common difference is 10, and the number of terms is 60; what is their sum? Ans. 18000. 6. The first term is 116, the common difference is -4, and the number of terms is 25; what is their sum? 7. The first term is 1, the common difference is 4, and the number of terms is 12; what is their sum? 8. The first term is 13, the common difference is -; and the number of terms is 10; what is their sum? 9. Required the number of terms of a progression whose sum is 442, whose first term is 2, and common difference 3. Ans. 17. 10. Required the first term of a progression whose sum is 99, whose last term is 19, and common difference 2. 11. The sum of a progression is 1455, the first term 5, and the last term 92; what is the common difference? 12. Required the sum of 101 terms of the series 1, 3, 5, 7, 9, etc. 13. Find the nth term of the series 1, 3, 5, 7, 9, etc. 14. Find the sum of n terms of the series 1, 3, 5, 7, 9, etc. Ans. 10201. Ans. 2n-1. Ans. n2. 15. Find the sum of n terms of the series of numbers 1, 2, 3, 4, 5, etc. 16. Find the sum of n terms of the series 2, 4, 6, 8, etc. Ans. n(n+1). and 3. 17. Find 6 arithmetical means between 1 and 50. 18. Find 7 arithmetical means between 19. A body falls 16 feet during the first second, and in each succeeding second 32 feet more than in the one immediately preceding; if it continue falling for 20 seconds, how many feet will it pass over in the last second, and how many in the whole time? Ans. 624 feet in the last second, and 6400 feet in the whole time. 20. 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 which is placed two yards from the first stone? Ans. 20200 yards. PROBLEMS. 327. When of the five quantities a, l, n, d, s, no three are directly given, it may be necessary to represent the series by the use of two unknown quantities. The form of the series which will be found most convenient will depend upon the conditions of the problem. If x denote the first term and y the common difference, then x, x+y, x+2y, x+3y, etc., will represent a series in arithmetical progression. It will, however, generally be found most convenient to represent the series in such a manner that the common difference may disappear in taking the sum of the terms. Thus a progression of three terms may be represented by x-y, x, x+y; one of four terms by x-3y, x-y, x+y, x+3y; one of five terms by x-2y, x-y, x, x+y, x+2y. Prob. 1. A number consisting of three digits which are in arithmetical progression, being divided by the sum of its digits, gives a quotient 26; and if 198 be added to it, the digits will be inverted; required the number. Ans. 234. Prob. 2. Find three numbers in arithmetical progression the sum of whose squares shall be 1232, and the square of the mean greater than the product of the two extremes by 16. Ans. 16, 20, and 24. Prob. 3. Find three numbers in arithmetical progression the sum of whose squares shall be a, and the square of the mean greater than the product of the two extremes by b. ; and a-26 + √b. Prob. 4. Find four numbers in arithmetical progression whose sum is 28, and continued product 585. Ans. 1, 5, 9, 13. Prob. 5. A sets out for a certain place, and travels 1 mile the first day, 2 the second, 3 the third, and so on. In five days afterward B sets out, and travels 12 miles a day. will A travel before he is overtaken by B? How long Ans. 8 or 15 days. This is another example of an equation of the second degree, in which the two roots are both positive. The following diagram exhibits the daily progress of each traveler. The divisions above the horizontal line represent the distances traveled each day by A; those below the line the distances traveled by B. A. 123 4 5 6 7 8 9 10 11 12 13 14 15 It is readily seen from the figure that A is in advance of B until the end of his 8th day, when B overtakes and passes him. After the 12th day, A gains upon B, and passes him on the 15th day, after which he is continually gaining upon B, and could not be again overtaken. Prob. 6. A goes 1 mile the first day, 2 the second, and so on. B starts a days later, and travels b miles per day. How long will A travel before he is overtaken by B? In what case would B never overtake A? (2b-1)2 Ans. When a> 86 For instance, in the preceding example, if B had started one day later, he could never have overtaken A. Prob. 7. A traveler set out from a certain place and went 1 mile the first day, 3 the second, 5 the third, and so on. After he had been gone three days, a second traveler sets out, and goes 12 miles the first day, 13 the second, and so on. After how many days will they be together? Ans. In 2 or 9 days. Let the student illustrate this example by a diagram like the preceding. Prob. 8. A and B, 165 miles distant from each other, set out with a design to meet. A travels 1 mile the first day, 2 the second, 3 the third, and so on. B travels 20 miles the first day, 18 the second, 16 the third, and so on. In how many days will they meet? Ans. 10 or 33 days. GEOMETRICAL PROGRESSION. 328. A geometrical progression is a series of quantities each of which is equal to the product of the preceding one by a constant factor. The constant factor is called the ratio of the series. 329. When the first term is positive, and the ratio greater than unity, the series forms an increasing geometrical progression, as in which the ratio is 2. 2, 4, 8, 16, 32, etc., When the ratio is less than unity, the series forms a decreasing geometrical progression, as in which the ratio is 81, 27, 9, 3, etc., 330. In a geometrical progression having a finite number of terms, there are five quantities to be considered, viz., the first |