EXERCISES 174. 1. Find the sum of 20 terms of the series 99, 103, 107, 111, and .... Hence, l = a + (n − 1)d = 99 + 19 × 4 = 175, The required sum is 2740. 2. Find the sum of 25 terms of the series 8, 5, 2, 3. Find the sum of 18 terms of the series 11, 5. Find the sum of all the integers between excluding 50 and 75. 6. Find the sum of 15 terms in 11, 1, 2, 23, .... .... 50 and 75, 7. In formula (B) substitute for its value as given in formula (A) and derive a formula for S which does not contain l. 8. Find the sum of the first one hundred odd numbers. 9. Find the sum of the first one hundred even numbers. 10. How many of the integers 1, 2, yield the sum 55? n 3, ... must be added to Use the formula S = 2 (2 a + (n − 1)d), derived in Ex. 7. Do both answers satisfy the conditions of the problem? 11. How many terms of the series 5, 4, 3, . ... are necessary to yield a sum 9? Do both answers satisfy the conditions of the problem? 12. The second term of an arithmetical series is 11, the fifth term is 20; find the 14th term. 13. The first week a store was opened the expenses exceeded the income by $ 52.25. The second week the loss was $ 41.75. If the improvement in the trade continued at the same rate, how much profit was made in 24 weeks? 14. A man enters an office at a salary of $1200, which is increased annually $75. How much will the firm pay him during 18 years? 15. The three formulas found below give the salaries offered by three companies to men entering their employ. S is the monthly salary in dollars earned after a given number of years (n). Calculate which company after 20 years' employ will give the highest salary. How much does a man earn in 20 years in each case? (a) S = 100+1⁄2n, (b) S = 95 + 4 n, GEOMETRICAL SERIES (c) S = 88 +13 n. 175. The series 3, 6, 12, 24, is not arithmetical; the difference between successive terms is not the same. The successive terms are formed in accordance with a different law. It is readily seen that any term after the first is derived from the preceding one by multiplying by 2. Such a series is called a geometrical series or a geometrical progression. A geometric series is a succession of numbers in which each number after the first, when divided by the preceding number, always gives the same quotient. The quotient is called the common ratio. EXERCISES 176. Which of the following series are geometrical and which arithmetical? LAST TERM OF A GEOMETRICAL SERIES 177. If the first term of a geometrical series is a, the common ratio is r, the number of terms n, then an expression for 7, the last term, may be obtained by inspecting the terms in the general geometrical series, We observe that in the second term, the exponent of r is 1, in the third term it is 2, in the fourth term it is 3, in the fifth term it is 4. Evidently, in the nth term, the exponent of r is n-1. Hence we have the formula for the last term of a geometrical progression, = arn-1. EXERCISES (C) 178. 1. Find the seventh term in the geometrical series, 16, 32, 64, Here .... a = 16, r = 2, n = 7. Hence, 1 = arn-1 = 16(2)7-1 = 16 × 64 = 1024. The seventh term is 1024. This result may be verified easily by writing down the first seven terms of the series. .... 2. Find the 8th term of the series 5, 21, 11, .... 179. The geometrical means between two numbers are numbers which, together with the two given numbers as first and last terms, form a geometrical series. If the two given numbers are 8 and 1⁄2, then 4, 2, 1 are three geometrical means, because 8, 4, 2, 1, is a geometric series. 2 EXERCISES 180. 1. Insert four geometrical means between 5 and 160. The two given numbers and the four means make together 6 terms. We have a 5, 7 = 160, n = 6. = By (C), We must find r. 1 = arn−1, 160 = 5 r5. giữ = 32, Hence the required geometrical series is 5, 10, 20, 40, 80, 160. 2. Insert six geometrical means between 10 and 1280. Solve r7 = 128 by trial. 3. Insert four geometrical means between 3 a and 96 ao. 4. Insert one geometrical mean between 133 and 1197. What is the geometrical mean between two numbers? What is the arithmetical mean between two numbers? 5. Two numbers differ by 6, and their arithmetical mean exceeds their geometric mean by 1. Find the numbers. SUM OF A GEOMETRICAL SERIES 181. The sum of the first n terms of a geometrical series may be indicated thus, S = a + ar + ar2 + ... + arn¬3 + arn-2 + arn−1. (1) Multiply both sides of (1) by r, TS= = ar + ar2 + ar3 + + arn¬2 + arn−1 + arn. (2) Subtract (1) from (2) and observe that all the terms disappear in subtraction, except a and ar". We obtain, EXERCISES 182. 1. Find the sum of six terms of the geometrical series 11, 22, 44, .... Hence, Sα( — 1) — 11(26 — 1) _ 11 (64 − 1) r-1 = 2-1 = 1 2. Find the sum of seven terms of 1, 1, 1, 3. Find the sum of six terms of 1, 4. Find the sum of five terms of x, = 11 × 63 = 693. ... - √2, 2, - 2√2,.... - xy2, xy1, . . . 5. Show that formula (C) may be written S = For what values of r is this form more convenient than (C)? 6. Show that (C) may be written S = a - rl 7. What will $1000 amount to in four years, interest 3 %, compounded annually? 8. What will $500 amount to in two years at 4% annual interest, compounded semiannually? 9. What will $700 amount to in eight years at 4 % annual interest, compounded semiannually? We simplify this computation by the use of logarithms. The amount is given by the expression x = $700 (1.02)16. This answer is only approximate. Had we used a six-place table of logarithms, instead of the four-place table, the answer would have come out $960.95, correct to the nearest cent. |