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OPERATION.

= 3.

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

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.

it

Note. — When only a single mean is required to be inserted, found as in Art. 550; when only two, as in Art. 551.

may

be

EXAMPLES

2. Insert three geometrical means between 3 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 term by the Icast; involve the rario to a power cqual 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.

[blocks in formation]

22

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

3.090 900 3.106 225 3.121 600 3.152 500 3.183 600 3.214 900 4.183 627 4.214 943 4.246 464 4.310 125 4.374 616 4.439 943

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 11.463 879 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 14 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

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 46.995 828 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

40.709 634 42.759 060 47.084 214 54.669 126 63.705 766 74.483 823 28 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 38 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 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 96.848 629 110.012 382 142.993 339 187.507 577 247.776 496 44 89.048 409 101.238 331 115.412 877 151 143 006 199.758 032 266.120 851 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

27

TABLE,

SHOWING THE PRESENT WORTH OF AX AXNUITY OF ONE DOLLAR PER

ANNUM, TO CONTINUE FOR ANY NUMBER OF YEARS NOT EXCEEDING
FIFTY.

Years.

3 per cent. 34 per cent. 4 per cent.

5 per cent.

6 per cent. 7 per cent.

Years.

1

0.970 874 0.966 184 0.961 538 0.952 381 0.943 396 0.934 579 1 2 1.913 470 1.899 694 1.886 095 1.859 410 1.833 393, 1.808 017 2 3 2.828 611 2.801 637 2.775 091 2.723 248 2.673 012 2.624 314 3

3.717 098 3.673 079 3.629 895 3.545 951 3.465 106 3.387 209 4 5 4.579 707 4.515 052 4.451 822 4.329 477 4.212 364 4.100 195 5 6 5.417 191 5.328 553 5.242 137] 5.075 692 4.917 324 4.766 537

6.230 283 6.114 544 6.002 055 5.786 373 5.582 381 5.389 286 8 7.019 692 6.873 956' 6.732 745 6.463 213 6.209 744 5.971 295 8 9 7.786 109 7.607 687 7.435 332 7 107 822 6.801 692 6.515 228 9 10 8.530 203 8.316 605 8.110 896 7.721 735 7.360 087 7.023 577 10 11 9.252 624 9.001 551 8.760 477 8.306 414 7.886 875 7.498 669 11 12 9.954 004 9.663 334 9.385 074 8.863 252 8.383 844 7.942 671 12 13 10.634 955 10.302 738) 9.985 648\ 9.393 573 8.852 683 8.357 635 13 14 11.296 073 10.920 520 10.563 123 9.898 641 9.294 984 8.745 452 14 15 11.937 935 11.517 411 11.118 367 10.379 658 9.712 249 9.107 898, 15 16 12.561 102 12.094 117 11.652 296 10.837 770 10.105 895 9.446 632 16 17 13.166 118 12.651 321 12.165 669 11.274 066 10.477 260 9.763 206 17 18 13.753 513 13.189 682 12.659 297 11.689 587 10.827 603 10.059 070' 18 19 14.323 799 13.709 837 13.133 939 12.085 321 11.158 116 10.335 578 19 20 14.877 475 14.212 403 13.590 326 12.462 210 11.469 421 10.593 997 20 21 15.415 024 14.697 974 14.029 160 12.821 153 11.764 077 10.835 527 21 22 15.936 917 15.167 125 14.451 115 13.163 003 12.041 582 11.061 241 22 23 16.443 608 15.620 410 14.856 842 13.488 574 12.303 379 11.272 187 23 24 16.935 542 16.058 368 15.246 963 13.798 612 12.550 358 11.469 334 24 25 17.413 148 16.481 515 15.622 080 14.093 945 12.783 356 11.653 583 25 26 17.876 842 16.890 352 15.982 769 14.275 185 13.003 166 11.825 779 26 27 18.327 031 17.285 365 16.329 586 14.643 034 13.210 534 11.986 709 27 28 18.764 108 17.667 019 16.663 063 14.898 127 13,406 164 12.137 111, 28 29 19.188 455 18.035 767 16.983 715 15.141 074 13.590 721 12.277 674 29 30 19.600 441 18.392 045 17.292 033 15.372 451 13.764 831 12.409 041 30 31 20.000 428 18.736 276 17.588 494 15.592 811 13.929 086 12.531 814 31 32 20.338 766 19.068 865 17.873 552 15.802 677 14.084 043 12.646 555 32 33 20.765 792 19 390 208 18.147 646 16.002 549 14 230 230 12.753 790 33 34 21.131 837 19.700 684 18.411 198 16.192 204 14.368 141 12.854 009 34 35 21.487 220 20.000 661 18.664 613 16.374 194 14.498 246 12.947 672 35 36 21.832 252 20.290 494 18.908 282 16.546 852 14.620 987 13.035 208 36 37 22.167 235 20.570 525 19.142 579 16.711 287 14.736 780 13.117 017 37 38 22.492 462 20.841 087 19.367 864 16.867 893 14.846 019 13.193 473 38 39 22.808 215 21.102 500 19.584 485 17.017 041 14.949 075 18,264 928 39 40 23.114 772 21.355 072 19.792 774 17.159 086 15.046 297|13.331 709 40 41 23.412 400 21.599 104 19.993 052 17.294 368 15.138 016 13.394 120 41 42 23.701 359 21.834 883 20.185 627 17.423 208 15.224 54 13.452 449 42 43 23.981 902 22.062 689 20.370 795 17.545 912 15.306 173 13,506 962 43 44 24.254 274 22.282 791 20.548 841 17 662 773 15.383 182 13.557 908 44 45 24.518 71322.495 450 20.720 040 17.774 070'15.455 832 13.605 522 45 46 24.775 449 22.700 918 20.884 654 17.880 067 15.524 370 13.650 020 46 47 25.024 708 22.899 438 21.042 936 17.981 016 15.589 028 13.691 608 47 48 25.266 707 23.091 244 21.195 131 18.077 158 15.650 027 13.730 474 48 49 25.501 657 23.276 56+21.341 472 18.168 722 15.707 572 13.766 799 49 50 25 729 764 23.455 618'21.482 185 18.255 925 15.761 861 13.800 746' 50

OPERATION

Or,

574. To find the amount of an annuity, at compound interest, forborne, or in arrears, for any number of years.

Ex. 1. What will an annuity of $ 60, unpaid, or in arrears, 4 years, amount to, at 6 per cent. compound interest?

Ans. $ 262.476. The amounts of the several

payments form a geometrical 1.064 -1

series, of which the annuity is x 60 = 262.476. 1.06 1

the first term, the amount of

$ 1 for one year the ratio, the Or, 4.374616 X 60 262.476.

years the number of the terms, and the amount required is the

sum of the series. Hence, RULE. Find the sum of the series as in geometrical progression.

Multiply the amount of $ 1 for the given time, found in the table, by the annuity, and the product will be the requireil amount.

NOTE. — The amount of an annuity at simple interest corresponds to the sum of an arithmetical series, of which the annuity is the first term, the interest on the annuity for one term the common difference, and the time in years the number of terms.

2. What will an annuity of $ 500 amount to for 5 years, at 6 per cent. compound interest ?

Ans. $ 2818.546. 3. What is the amount of an annuity of $ 80, unpaid, or in arrears, for 9 years, at 5 per cent. compound interest ?

Ans. $ 882.125. 4. What is the amount of an annuity of $ 1000, forborne for 15 years, at 3} per cent. compound interest ?

Ans. $ 19295.68. 5. What will an annuity of $30, payable semiannually, amount to, in arrears for 3 years, at 7 per cent. compound interest?

6. Suppose a salary of $ 600 per year, payable quarterly, to remain unpaid 51 years; to what sum will it amount, at 6 per cent. compound interest ?

Ans. $ 3875.63. 575. To find the present worth of an annuity, at compound interest.

Ex. 1. What is the present worth of an annuity of $ 60, to be continued 4 years, at 6 per cent compound interest ?

Ans. $ 207.906.

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