OPERATION. Ex. 1. If the first term be 1, the ratio 3, and the number of terms 5, required the sum of the terms. Ans. 121. If we multiply the series 1, 3, 9, 27, (81 X 3) - 1 81 by the ratio 3, we shall obtain as a 121. 3 -1 second series 3, 9, 27, 81, 243, whose sum is three times the sum of the first series, and the difference between whose sum and the sum of the first series is evidently twice the sum of the first series. Now it will be observed that the two series bave their terms alike, with the exception of the first term in the first series, and the last in the second series. We have then only to subtract the first term in the first series from the last term in the second, and the remainder is twice the sum of the first series; and half this being taken gives the required sum of the series. Therefore the sum of the first series must be 242 • 2 121. RULE. Find the last term as in Art. 565, multiply it by the ratio, and the product less the first term divide by the ratio less 1; the result will be the sum of the series. NOTE 1. — If the ratio is less than a unit, the product of the last term mulplied by the ratio must be subtracted from the first term, and the remainder divided by unity or 1 decreased by the ratio. NOTE 2. When a descending series is continued to infinity, it becomes what is called an infinite series, whose last term must always be regarded as 0, and its ratio as a fraction. To find the sum of an infinite series, Divide the first term by 1 decreased by the fraction denoting the ratio, and the quotient will be the sum required. This process furnishes an expeditious way of finding the value of circulating decimals, since they are composed of numbers in geometrical progression, whose common ratios are to, išo, tdoo, &c. according to the number of factors contained in the repetend. Thus, .3333, &c. represents the geometrical series ið, To, Igðo, &c. whose first term is iš and common ratio io. EXAMPLES 2. The first term of a series is 5, the ratio ş, and the number of terms 6; required the sum of the series. (3) X 5 = 23; 263 X= ; 5 — 138 = 3:22; 333 = (1 - 3) = 3 = 13144, Ans. 3. Find the value of the circulating decimal .232323, &c. 12 + (1 - TOO) = 33, Ans. 4. What is the sum of the series 4, 1, 1, 16, &c., continued to an infinite number of terms ? 4 ; (1 - 1) = 51, Ans. 5. If the first term is 50, the ratio 1.06, and the number of terms 4, what is the sum of the series? Ans. 218.7308. 83 ون 6. A gentleman offered a house for sale on the following terms; that for the first door he should charge 10 cents, for the second 20 cents, for the third 40 cenis, and so on in a geometrical ratio. If there were 40 doors, what was the price of the house? Ans. $ 109951162777.50. 7. If the series 3, 4, ¿t, &c. were carried to infinity, what would be its sum? Ans. 11. 8. A gentleman deposited annually $ 10 in a bank, from the time his son was born until he was 20 years old. Required the amount of the 21 deposits at 6 per cent., compound interest, when his son was 21 years old ? Ans. $ 423.92. 9. Find the value of .008197133, &c., continued to infinity. Ans. 10. If a body be put in motion by a force which moves it 10 miles in the first portion of time, 9 miles in the second equal portion, and so, in the ratio of i, for ever, how many miles will it pass Ans. 100 miles. 567. To find the ratio, the extremes and number of terms being given. Ex. 1. If the extremes of a series are 3 and 192, and the number of terms 7, what is the ratio ? It has been shown (Art. 192 ; 3 64; ♡ 64 = 2, Ans. 565) that the last term of a geometrical series is equal to the product of the ratio by the first term raised to a power whose index is one less than the number of terms; hence the ratio must equal the root of the quotient of the last term by the first whose index is one less than the number of terms. Rule. — Divide the last extreme by the first, and extract that root of the quotient whose index is one less than the number of terms. NOTE. When the sum of the series and the extremes are given, the ratio may be found by dividing the sum of all the terms except the first, by the sum of all the terms except the last. EXAMPLES. 2. If the last term of a series is 1, the largest term 512, and the number of terms 10; what is the ratio ? Ans. 2. 3. If the extremes are 5 and 885735, and the sum of the series 1328600, what is the ratio ? Ans. 3. 4. What debt can be discharged in a year by monthly pay OPERATION. OPERATION. ments, in geometrical progression, of which the first payment is $ 1, and the last $ 2048; and what will be the ratio of the series ? Ans. Ratio, 2; debt, $ 4095. 568. To insert any number of geometrical means, or mean proportionals, between two given numbers. Ex. 1. Insert three geometrical means between 4 and 324. Ans. 12, 36, and 108. Since the series will in324 + 4 = 81; 81 = 3. clude, beside 4 X 3 = 12; 12 X 3 = 36; 36 X 3 = 108. the inserted terms, the two extremes, the number of terms will be 5. Then, having the number of terms and the extremes, we find the ratio, as in the last article, to be 3; and by multiplying the first term by the ratio, we obtain the first of the terms to be inserted. That term multiplied by the ratio gives the next, and that multiplied by the ratio gives the other required mean. RULE.— Take the two given numbers as the extremes of a geometrical series, and consider the number of terms in the series greater by two than the required number of means. Then find the ratio, as in Art. 567, and the product of the ratio and the first extreme will give one of the means, and the product of this mean and the ratio will give another, and 80 on, NOTE. — When only a single mean is required to be inserted, it may be found as in Art. 550; when only two, as in Art. 551. EXAMPLES 2. Insert three geometrical means between į and 128. Ans. 2, 8, and 32. 3. Required five mean proportionals between the numbers 3 and 2187. Ans. 9, 27, 81, 243, and 729. 569. To find the number of terms, the extremes and ratio being given. Ex. 1. If the extremes are 5 and 3645, and the ratio 3, what is the number of terms ? By Art. 565 it is seen that the ratio 3645 ; 729; 36 = 729. raised to the power whose index is one 6+1= 7, Ans. less than the number of terms, and multiplied by the least term, equals the largest term; hence, the largest term divided by the least term will equal a power of the ratio whose index is one less than the number of terms. OPERATION. RULE. Divide the largest lerm by the least; involve the ratio to a power equal to the quotient, and the index of that power, increased by 1, will be the number of terms. EXAMPLES 2. If the extremes are 5 and 20480, and the ratio 4, what is the number of terms ? Ans. 7. 3. In what time will a certain debt be discharged by monthly payments in geometrical progression, if the first and last payments are $ 1 and $ 2048, and the ratio 2 ? Ans. In 12 months. ANNUITIES. 570. ANNUITIES are fixed sums of money payable at the ends of equal periods of time, such as years, or half-years. Annuities in perpetuity are such as continue for ever. Annuities certain are such as commence at a fixed time, and continue for a certain number of years. Annuities contingent are those whose commencement or continuance, or both, depend on some contingent event, as the death of one or more individuals. Annuities deferred, or in reversion, are such as do not commence till after a fixed number of years, or till after some particular event has taken place. 571. An annuity forborne, or in arrears, is one whose periodical payments, instead of being paid when due, have been allowed to accumulate. 572. The amount of an annuity at compound interest, at any time, is the sum to which it will amount, supposing it to have been improved at compound interest during the intervening period. 573. The present value of an annuity at compound interest, for any given period, is the sum of the present values of all the payments of that annuity. TABLE, SHOWING THE AMOUNT OF AN ANNUITY OF ONE DOLLAR PER ANNUM, IMPROVED AT COMPOUND INTEREST FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY. 14 1 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000 2 2.030 000 2.035 000 2.040 000 2.050 000 2.060 000 2.070 000 3 3.090 900 3.106 225 3.121 600 3.152 500 3.183 600 3.214 900 4 4.183 627 4.214 943 4.246 464 4.310 125 4.374 616 4.439 943 5 5.309 136 5.362 466 5.416 323 5.525 631 5.637 093 5.750 739 6 6.468 410 6.550 152 6.632 975 6.801 913 6.975 319 7.153 291 7 7.662 462 7.779 408 7.898 294 8.142 008 8.393 838 8.654 021 8 8.892 336 9.051 687 9.214 226 9.549 109 9.897 468 10.259 803 9 10.159 106 10.368 496 10.582 795 11 026 564 11.491 316 11.977 989 10 | 11.463 979 11.731 393 12.006 107 12.577 893 13.180 795 13.816 448 11 12.807 796 13.141 992 13.486 351 14.206 787 14.971 643 15.783 599 12 14.192 030 14.601 962 15.025 805 15.917 127 16.869 941 17.888 451 13 15.617 790 16.113 030 16.626 838 17.712 983 18.882 138 20.140 643 17.086 324 17.676 986 18.291 911 19.598 632 21.015 066 22.550 488 15 18.598 914 19.295 681 20.023 588/ 21.578 564 23.275 970 25.129 022 16 20.156 881 20.971 030 21.824 531 23.657 492 25.670 528 27.888 054 17 21.761 588 22.705 016 23.697 512 25.840 366 28.212 880 30.840 215 18 23.414 435 24.499 691 25.645 413 28.132 385 30.905 653 33.999 033 19 25.116 868 26.357 180 27.671 229 30.539 004 33.759 992) 37.378 965 20 26.870 374 28.279 682 29.778 079 33.065 954 36.785 591 40.995 492 21 28.676 486) 30.269 471 31.969 202 35.719 252 39.992 727 44.865 177 22 30.536 780 32.328 902 34.247 970 38.505 214 43.392 290 49.005 739 23 32.452 884 34.460 414 36.617 889 41.430 475 46.995 8281 53.436 141 24 34.426 470 36.666 528 39.082 604 44.501 999 50.815 577 58.176 671 25 36.459 264 38.949 857 41.645 908 47.727 099 54.864 512 63.249 030 26 38.553 042 41.313 102 44.311 745 51.113 454 59.156 383 68.676 470 27 40.709 634 42.759 060 47.084 214 54.669 126 63.705 766 74.483 823 42.930 923' 46.290 627 49.967 583 58.402 583 68.528 112 80.697 691 29 45.218 850 48.910 799 52.966 286 62.322 712 73.639 798 87.346 529 30 47.575 416 51.622 677 56.084 938 66.438 848 79.058 186 94.460 786 31 50.002 678 54.429 471 59.328 335 70.760 790 84.801 677 102.073 041 32 52.502 759 57.334 502 62.701 469 75.298 829 90.889 778 110.218 154 33 55.077 841 60.341 210 66.209 527 80.063 771 97.343 165 118.933 425 34 57.730 177 63.453 152 69.857 909 85.066 959 104.183 755 128.258 765 35 60.462 082 66.674 013 73 652 225 90.320 307 111.434 780 138.236 878 36 63 271 944 70.007 603 77.598 314 95.836 323 119.120 867 148.913 460 37 66.174 223 73.457 869 81.702 246 101.628 139 127.268 119 160.337 400 69.159 449 77.028 895 85.970 336 107.709 546 135.904 206 172.561 020 39 72.234 233 80.724 906 90.409 150 114.095 023'145.058 458 185.640 292 1 40 75.401 260 84.550 278 95.025 516 120.799 774 154.761 966 199.635 112 41 78.663 298 88.509 537 99.826 536 127.839 763 165.047 684 214.609 570 42 82.023 196 92.607 371 104.819 598 135.231 751 175.950 645 230.632 240 43 85.483 892 96.848 629 110.012 382 142.993 359 187.507 577 247.776 496 44 89.048 409 101.238 331 115.412 877151 143 006'199.758 032 266.120 851 45 92.719 861 105.781 673 121.029 392 159.700 156 212.743 514 285.749 311 46 96.501 457 110.484 031 126.870 568 168.685 164.226.508 125 306.751 763 47 100.396 501 115.350 973 132.945 390 178.119 422 241.098 612 329.224 386 48 104.408 396 120.388 297 139.263 206 188.025 393 256.564 529 353.270 093 49 108.540 648 125.601 846 145.833 734 198.426 663 272.958 401 378.999 000 50 112.796 867 130.999 910 152.667 084 209.347 976 290.335 905 406.528 929 28 38 |