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.

OPERATION.

3244= 81; 81 = 3.

4 × 3 = 12; 12 × 3 = 36; 36 X 3108.

Since the series will in

clude, beside 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

so 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

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?

OPERATION.

3645729; 36

617, Ans.

729.

By Art. 565 it is seen that the ratio raised to the power whose index is one 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.

RULE. · Divide the largest term 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.

32

52.502 759

57.834 502

33 55.077 841

60.341 210

34

35

57.730 177
60.462 082

36

63.453 152
66.674 013
70.007 603
73.457 869
77.028 895
80.724 906
84.550 278
88.509 537

90.889 778 110.218 154

97.343 165 118.933 425

62.701 469
66.209 527
69.857 909 85.066 959 104.183 755 128.258 765
73.652 225 90.320 307 111.434 780 138.236 878
63 271 944
77.598 314 95.836 323 119.120 867 148.913 460
37 66.174 223
81.702 246 101.628 139 127.268 119 160.337 400
38 69.159 449
85.970 336 107.709 546 135.904 206 172.561 020
39 72.234 233
90.409 150 114.095 023 145.058 458 185.640 292
40 75.401 260
95.025 516 120.799 774 154.761 966 199.635 112
41 78.663 298
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

70.760 790
75.298 829
80.063 771

84.801 677 102.073 041

TABLE,

SHOWING THE PRESENT WORTH OF AN ANNUITY OF ONE DOLLAR PER ANNUM, TO CONTINUE FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY.

8.863 252

8.383 844

7.942 671 12

9.393 573

8.852 683

8.357 635 13

9.294 984

12 9.954 004 9.663 334 9.385 074 13 10.634 955 10.302 738 9.985 648 14 11.296 073 10.920 520 10.563 123 9.898 641 8.745 452 14 15 11.937 935 11.517 411 11.118 387 10.379 658 9.712 249 9.107 898 15 296 10.837 770 10.105 895 9.446 632 16 669 11.274 066 10.477 260 9.763 206 17 297 11.689 587 10.827 603 10.059 070 18 939 12.085 321 11.158 116 10.335 578 19 326 12.462 210 11.469 421 10.593 997 20 160 12.821 153 11.764 077 10.835 527 21 115 13.163 003 12.041 582 11.061 241 22 842 13.488 574 12.303 379 11.272 187 23 963 13.798 642 12.550 358 11.469 334 24 080 14.093 945 12.783 356 11.653 583 25 769 14.275 185 13.003 166 11.825 779 26 586 14.643 034 13.210 534 11.986 709 27 063 14.898 127 13.406 164 12.137 111 28 715 15.141 074 13.590 721 12.277 674 29 033 15.372 451 13.764 831 12.409 041 30 494 15.592 811 13.929 086 12.531 814 31 552 15.802 677 14.084 043 12.646 555 32 646 16.002 549 14.230 230 12.753 790 33 198 16.192 204 14.368 141 12.854 009 34 613 16.374 194 14.498 246 12.947 672 35 282 16.546 852 14.620 987 13.035 208 36 579 16.711 287 14.736 780 13.117 017 37 893 14.846 019 13.193 473 38 041 14.949 075 13.264 928 39 086 15.046 297 13.331 709 40 368 15.138 016 13.394 120 41 208 15.224 543 13.452 449 42 912 15.306 173 13.506 962 43 773 15.383 182 13.557 908 44 070 15.455

16 12.561 102 12.094 117 11.652 17 13.166 118 12.651 321 12.165 18 13.753 513 13.189 682 12.659 19 14.323 799 13.709 837 13.133 20 14.877 475 14.212 403 13.590 21 15.415 024 14.697 974 14.029 22 15.936 917 15.167 125 14.451 23 16.443 608 15.620 410 14.856 24 16.935 542 16.058 368 15.246 25 17.413 148 16.481 515 15.622 26 17.876 842 16.890 352 15.982 27 18.327 031 17.285 365 16.329 28 18.764 108 17.667 019 16.663 29 19.188 455 18.035 767 16.983 30 19.600 441 18.392 045 17.292 31 20.000 428 18.736 276 17.588 32 20.338 766 19.068 865 17.873 33 20.765 792 19 390 208 18.147 34 21.131 837 19.700 684 18.411 35 21.487 220 20.000 661 18.664 36 21.832 252 20.290 494 18.908 37 22.167 235 20.570 525 19.142 38 22.492 462 20.841 087 19.367 39 22.808 215 21.102 500 19.584 40 23.114 772 21.355 41 23.412 400 21.599 42 23.701 359 21.834 43 23.981 902 22.062 689 20.370 44 24.254 274 22.282 791 20.548 45 24.518 713 22.495 450 20.720 46 24.775 449 22.700 918 20.884 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 23.276 564 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

864 16.867

485 17.017

072 19.792

774 17.159

104 19.993

052 17.294

883 20.185

627 17.423

795 17.545

841 17.662

040 17.774

654 17.880

832 13.605 522 45 067 15.524 370 13.650 020 46

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

4

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

Ans. $262.476.

The amounts of the several payments form a geometrical series, of which the annuity is the first term, the amount of $1 for one year the ratio, the 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.

Or,

Multiply the amount of $1 for the given time, found in the table, by the annuity, and the product will be the required 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 5 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.