EXERCISE 127. 1. Find the seventh term of 2, 6, 18.. 2. Find the sixth term of 3, 6, 12 ..... 3. Find the ninth term of 6, 3, 11⁄2 .... 4. Find the eighth term of 1, — 2, 4 · ..... 5. Find the fifth term of 4 a, — 6 ma2, 9 m2as .... 6. Find the geometrical mean between 18x3y and 30xy3z. 7. Find the ratio when the first and third terms are 5 and 80, respectively. 8. Insert two geometrical means between 8 and 125; and three between 14 and 224. 9. If a 2 and r = 3, which term will be equal to 162 ? 10. The fifth term of a geometrical series is 48, and the Find the first and seventh terms. ratio 2. 16. The population of a city increased in four years from 10,000 to 14,641. What was the annual rate of increase? 17. The sum of four numbers in geometrical progression is 200, and the first term is 5. Find the ratio. 18. Find the sum of eight terms of a geometrical series whose last term is 1, and fifth term }. 19. In an odd number of terms, show that the product of the first and last is equal to the square of the middle term. 20. The product of four terms of a geometrical series is 4, and the fourth term is 4. Find the series. 21. If from a line one third is cut off, then one third of the remainder, and so on, what fraction of the whole will remain when this has been done five times? 22. Of three numbers in geometrical progression, the sum of the first and second exceeds the third by 3, and the sum of the first and third exceeds the second by 21. are the numbers? What 23. Find two numbers whose sum is 34 and geometrical mean 11. 24. The sum of the squares of two numbers exceeds twice their product by 576; the arithmetical mean of the two numbers exceeds the geometrical mean by 6. Find the numbers. 25. There are four numbers such that the sum of the first and the last is 11, and the sum of the other two is 10. The first three of these four numbers are in arithmetical progression, and the last three are in geometrical progression. Find the numbers. 26. Find three numbers in geometrical progression such that their sum is 13 and the sum of their squares 91. 27. The difference between two numbers is 48, and the arithmetical mean exceeds the geometrical mean by 18. Find the numbers. 28. There are four numbers in geometrical progression, the second of which is less than the fourth by 24, and the sum of the extremes is to the sum of the means as 7 to 3. Find the numbers. Infinite Geometrical Series. 370. When r is less than 1, the successive terms of a geometrical series become numerically smaller; by taking n large enough we can make the nth term, ar"-1, as small as we please, although we cannot make it absolutely zero. sidered the sum of an infinite number of terms of the series. 1. Find the sum of 1 − 1 + 1 − } + ··· The terms after the first form a decreasing geometrical series in which a = 0.036, and r = 0.01. * Harmonical Progression. 371. A series is called a harmonical series or a harmonical progression when the reciprocals of its terms form an arithmetical series. Hence, the general representative of such a series is 372. Questions relating to harmonical series should be solved by writing the reciprocals of its terms so as to form an arithmetical series. 373. If a and b denote two numbers, and H their harmonical mean, then, by the definition of a harmonical series, 374. Sometimes it is required to insert several harmonical means between two numbers. Let it be required to insert three harmonical means between 3 and 18. Find the three arithmetical means between 14 9 and. These are found to be 12, 44, 72; therefore, the harmonical means are 13, 14, 2; or 315, 5, 8. *A harmonical series is so called because musical strings of uniform thickness and tension produce harmony when their lengths are represented by the reciprocals of the natural series of numbers; that is, by the harmonical series 1,,, 1, 1, etc. EXERCISE 129. 1. Insert four harmonical means between 2 and 12. 2. Find two numbers whose difference is 8 and harmonical mean 1. 3. Find the seventh term of the harmonical series 3, 33, 4 4. Continue to two terms each way the harmonical series two consecutive terms of which are 15, 16. 6. 5. The first two terms of a harmonical series are 5 and Which term will equal 30? 6. The fifth and ninth terms of a harmonical series are 8 and 12. Find the first four terms. 7. The difference between the arithmetical and harmonical means between two numbers is 14, and one of the numbers is four times the other. Find the numbers. 8. Find the arithmetical, geometrical, and harmonical means between two numbers, a and b; and show that the geometrical mean is a mean proportional between the arithmetical and harmonical means. Also, arrange these means in order of magnitude. 9. The arithmetical mean between two numbers exceeds the geometrical by 13, and the geometrical exceeds the harmonical by 12. What are the numbers? 10. The sum of three terms of a harmonical series is 11, and the sum of their squares is 49. Find the numbers. 11. When a, b, c are in harmonical progression, show that a ca b: b C. |