whose common difference is d: that is, the arithmetical mean of the a+d, a +2d. series a, 349. The two formulas, l=a+(n-1)d, s = (a+1)n, (1) (2) are fundamental, since from them may be derived other formulas, by which, when any three of the five elements of an arithmetical progression are given, the other two can be determined. Thus, we may obtain, among others: 3. Given a = 7, 7=31 and n = 9, to find S. 4. Given a = 10, 770 and n = 21, to find d. 5. Given l=31, d=3 and n = 9, to find a. Ans. S-171. Ans. a = 7. Ans. n = 9. to 20 terms. 9. Insert five arithmetical means between 11 and 23. SOLUTION. Here, l=23, a=11 and m= =5; whence, by formula (4), Art. 347, d=2. Hence, by addition, the progression will be 11, 13, 15, 17, 19, 21, 23. 10. Find an arithmetical mean between 12 and 20. Ans. 16. 11. Form an arithmetical progression by inserting 4 arithmetical means between 2 and -18. 12. How many strokes does a common clock make in 12 hours, if it strikes the half hours? Ans. 90. 13. A debt can be discharged in a year by paying $1 the first week, $3 the second, $5 the third, and so on. Required the last payment and the amount of the debt. 14. A person has a journey of 140 miles to perform. He goes 26 miles the first day, 24 miles the second, 22 the third, and so on. In how many days does he perform the journey? SECTION XLIV. GEOMETRICAL PROGRESSION. 350. A Geometrical Progression is a series in which each term, after the first, is equal to the preceding term multiplied by a constant factor. 351. The Rate or Ratio is the constant factor. The progression is ascending when the rate is greater than 1, and descending when the rate is less than 1. Thus, a, ar, ar2, ars.... is a geometrical progression in which r is the rate, and which is ascending or descending, according as r is greater or less than 1. If the rate is a negative quantity, the terms of the progression will be alternately positive and negative, or negative and positive, according as the first term is positive or negative. -- Thus, if the rate is 2, and the first term is 5, the progression will be, and if the first term is -5, the progression will be, 352. An Infinite Series is a descending progression of an infinite number of terms. The last term of such a series must be less than any assignable quantity; hence, it may be considered to be 0. Thus, 1, 1, O is an infinite series. 353. The Sum of an infinite series, or the sum of the series to infinity, is the limit which the sum approaches as the number of terms increases. The sum of the series 1, 1, 1, ... is constantly approaching the limit 2, and can never go beyond it. Hence, if the series be continued to infinity, its sum will be 2. 354. In a Geometrical Progression there are five elements to The first and last terms are called Extremes, and the other terms are called Geometrical Means. Theorem I. 355. The last term of a geometrical progression is equal to the product of the first term by that power of the rate whose exponent is one less than the number of terms. Let the terms of the progression be a, ar, ar2, ar3... The exponent of r in the last term is one less than the number of terms; hence, the nth term equals arn-1 356. The sum of all the terms of a geometrical progression is equal to the difference between the first term and the product of the last term, multiplied by the rate, divided by the rate less one. Let a geometrical progression be a, ar, ar2, 357. The sum of an infinite series is equal to the quotient of the first term divided by one minus the rate. The progression being descending, formula (2), in order that the denominator may be positive, must become, Now, since the number of terms in the descending series is infinite, the last term may be considered 0, and the last term multiplied by the rate will also be 0. Hence, the formula (3) will become, 358. Any number of geometrical means may be inserted between two given terms of a geometrical progression. The number of terms in a series must consist of two more terms than the number of means. Hence, if m be made to denote the number of means, m+2=n, the whole number of terms. Substituting this value of n in formula (1), Art. 355, we have, Having found the rate, we may obtain the required means by successively multiplying the first term by the rate, by its square, by its cube, etc. Theorem V. 359. The geometrical mean between two quantities is the square root of their prodact. But ar is the second term of a series whose first term is a, and whose rate is r; that is, the geometrical mean of the series a, αν, ar2 |