SERIES OF EQUAL RATIOS. 1 5 6 A Series of Equal Ratios, improperly called Geometrical Progression, is one in which the ratios of any two consecutive terms taken in the same order is equal; as, 1, 2, 4, 8, 16, is an increasing series, and any term divided by the one immediately preceding it gives a quotient of 2. 2 4 Observe the series, 1, 2, 4, 8, 16, 32. Write it literally, O, ar?, ars, art, ar5. a = 1st term, and r = ratio. Each term is equal to the product of the 1st term and the ratio raised to the power of the number of terms less one. FORMULA (1). nth term = arn-l. ar, PROBLEM. (1) To find the sum of a series of equal ratios. S = a +ar+ar+ar+art +ars, and multiply both terms by r, Sr = ar+arl + ar3+ art + apb + arb. Subtract the 1st equation from the 2d, Sr - S = arb - a = a (76 — 1). (S = (2) 9. The first term of a series is 1 and the number of terms 23; what must be the common difference in order that the sum may be 1494 ? Com. diff. = : = FORMULA 2. 1= a + (n − 1) d. Subtract a from both members of Formula 2 and divide by (n − 1), Į - (n − 1) d. 13 = . Ans. a = a d= n The common difference is equal to the last term minus the first divided by the number of terms less one. REM.—When terms are to be inserted, find the common difference and then apply it. 10. What is the sum of an increasing series whose first term is }, the common difference 1, and the number of the terms 20 ? Sum = 105. 11. What is the sum of an increasing series whose first term is 2, common difference 3, and number of terms 12 ? S= 222. COMPOUND INTEREST. Compound Interest consists in adding the interest to the principal as often as the interest becomes due, until the end of the time at which it is at interest. When a sum of money is at interest for a considerable time, and no interest has been collected, although it was specified that the interest was payable annually, semiannually, or quarterly, and the interest is computed on the principal for the specified time of interest, and at the end of such period added to the principal, this is called Compound Interest; thus, the interest of $1 at 6% for 1 year is $.06, which if added to the principal makes $1.06. The ratio will be 106, or 1.06; therefore, $1.06 is the amt. of $1 for 1 yr. at 6% = R. 1.06 1.1236 2 yrs. at 6% = RP. 1.06 1.191016 3 yrs. at 6% = Rs. 4 yrs. at 6% 5 yrs. at 6% 6 yrs. 1 .418519, called 42 cents, is the compound interest of $1 for 6 years at 6%. = R4 = Rs. at 6% = R6. Let R = 1.06, then R?, etc. will be the amt. of $1 for one, two, etc. years, R" for n years. If the interest were payable semi-annually, it would be 1.03 to the power expressed by double the number of years; thus, 1.03 = R. (1.03)2 = R12. If quarterly, (1.015)24 = R4 12 R representing respectively 1.06, 1.03, and 1.015. After the manner of the above computation, form a table for the amount of $1 for 50 years at 2%, 25%, etc., to 10%, and as a number in being multiplied by itself is said to be raised to higher powers, as above, When a number is raised to higher powers, the product has the sum of the powers; therefore, by this table, the compound amount of $1 for any number of years can easily be found, and from the amount subtract 1 and the remainder will be the compound interest. REM.-If there be a few months or days time above the speci. fied time which cannot be had from the tables, take the highest amount in the tables and compute the balance of time on it as in simple interest, and add it to the amount obtained from the tables, TABLE showing the amount of $1, at 21, 3, 3, 4, 5, and 6%, compound int., from 1 to 20 years. 1 1.025000 1.030000 1.035000 1.040000 1.050000 (1.060000 2 1.050625 1.060900 1.071225 1.081600 1.102500 1.123600 3 1.076891 1.092727 | 1.108718 1.124864 1.157625 1.191016 4 1.103813 1.125509 1.147523 1.169859 1.215506 1.262477 5 1.131408 1.159274 1.187686 1.216653 1.276282 1.338226 6 1.159693 1.194052 1.229255 1.265319 1.340096 1.418519 n 1.188686 1.229874 1.272279 1.315932 1.407100 1.503030 8 1.218403 1.266770 1.316809 1.368569 1.477455 1.593848 9 1.248863 1.304773 1.362897 1.423312 1.551328 1.689479 10 1.280085 1.343916 1.410599 1.480244 1.628885 1.790818 11 1.312087 1.384234 1.459970 1.539454 1.7103:39 1.898299 12 1.344889 1.425761 1.511069 1.601032 1.795856 2.012197 13 1.378511 | 1.468534 1.563956 1.665074 1.885649 2.132928 14 1.412974 1.512590 1.618695 1.731676 1.979932 2.260904 15 1.448298 1.557967 1.675349 1.800944 2.078928 2.396558 16 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352 17 1.521618 1.652848 1.7946761.947901 2:292018 2.692773 18 1.559659 1.702433 1.857489 2.025817 2.406619 2.854339 19 1.598650 1.753506 1.922501 2.106849 2.526950 3.025600 20 1.638616 1.806111 1.989789 2.191123 2.653298 3.207136 EXAMPLES. 1. What is the compound amount of $500 for 12 yrs., at 6% ? The number in the table opposite 12 years, at 6%, is 2.0121965 x 500 = $1006.09825. If for 12 years and 6 months, 2.0121965 x 1.03 = 2.0725624 x 500 = $1036.2812. If for 12 years and 3 months, x 1.015. |