MULTIPLY THE GREATEST TERM BY THE RATIO, FROM THE PRODUCT SUBTRACT THE LEAST TERM, AND DIVIDE THE REMAINDER BY THE RATIO, LESS 1. 2. Given the first term, 1; the last term, 2,187; and the ratio, 3: required the sum of the series. A. 3,280 3. Extremes, 1 and 65,536; ratio 4: required the sum of the scries. A. 87,381 4. Extremes 1,024 and 59,049: required as above. A. 175,099. 5. What is the sum of the series 16, 4, 1, 4, T6, TA, and so on, to an infinite extent ? A. 213. NOTE. The last term is evidently 0. 6. A man had a debt to pay as follows: the first month $3; the second, $6; the third, $12, and so on, in a twofold ratio. In 12 months the debt was paid. What was the last payment? The second payment is found by multiplying the first by the ratio, 2, once; the third, by multiplying by 2, twice; and so on, where it will be seen that the ratio is always used as a multiplier, a single time less than the number of terms. Hence, the 12th. term the first termXthe ratio eleven times successively, or, in other words, the eleventh power of the ratio, thus, 3×2×2×2×2×2×2×2×2×2×2×2=3×211=6,144 Ans. Hence, the first term, ratio, and number of terms being given, to find the last term, MULTIPLY THE FIRST TERM BY THAT POWER OF THE RATIO, WHOSE INDEX IS 1 LESS THAN THE NUMBER OF TERMS. NOTE. In involving the ratio, it will be seen that the process may often be abridged by multiplying together two powers already obtained. Thus, the 3d power the 2d power the 5th power, &c. 7. If the first term is 2, the ratio, 2, and the number of terms, 13, what is the last term? A. 8,192. 8, Find the 12th. term of a series, whose 1st. term is 3, and ratio, 3. A. 531,441. 9. First term, 1, ratio 2; required the 23d. term. A. 4,194,304. 10. A man bought a horse, giving 1 ct. for the first nail in his shoes, three for the second, and so on; there were 32 nails; what cost the horse ? A. 9,265,100,944,259.20. 11. A man works for a farmer 40 years, receiving 1 kernel of corn for the first year, 10 for the second, and so on; what do his wages amount to, allowing 1,000 kernels to a pt., and supposing corn worth 50 cts. per bu.? A. $8,680,555,555,555,555,555,555,555,555,555,555.555555 12. A young man agreed to work eleven years with a farmer, on condition of receiving the produce of one wheat corn, the first year; the produce of that quantity, sowed the second year; and so on, to the end of the time. How much wheat was there due for his service, and what would it come to at $1 per bu., allowing the yearly increase to have been tenfold, and 7,680 corns to make a pt.? A. 226,056 bu. at $226,056.125 NOTE. The cases of Geometrical Progression are numerous. Ten different ones may be stated, with two requisitions in each case, making, in fact, 20 different cases. The same is true in Arithmetical Progression. It would require much space to give a clear understanding of all these cases, and as they are of little practical importance, we pass them by. The terms Arithmetical and Geometrical, are used without regard to their proper signification, (viz. belonging to Arithme. tic and to Geometry,) but only to distinguish these different kinds of Progression. ANNUITIES. § CIV. 1. A pension is allowed a man and his heirs, forever, of $600. He is willing to dispose of this pension at a fair price. What ought he to receive for it, allowing money to draw 6 per ct. interest? It is evident he ought to receive a sum which would produce an annual interest, equal to the pension. $600 then is the interest, and the principal is acquired at 6 pr. ct. Hence, (§ LXXXI.) $600÷.06= $10,000 Ans. A sum of money, payable periodically, for a certain length of time, or forever, is called an ANNUITY. An annuity, in the proper sense of the word is a sum, payable annually. Payments, made at greater or less periods, are, however called annuities. To annuities belong pensions, salaries, rents, &c. When an annuity remains unpaid after it is due, it is said to be in arrears. The sum of the annuities in arrears, with the interest on each, is called the AMOUNT. The sum, which ought to be paid for an annuity yet to called the PRESENT WORTH. come, is From the above example, it is evident that, to find the present worth of an annuity to continue forever, we must DIVIDE THE ANNUITY BY THE RATE PER CENT. 2. What should be paid for a perpetual annuity of $40, discounting at 5 pr. ct.? Ans. 800. 3. What is an estate worth, which brings in $7,500 a year allowing 6 pr. ct.? Ans. $125,000. ANNUITIES AT SIMPLE INTEREST. 1. If a rent of $600 be in arrears 5 years, what will be due at 4 per cent.? On the rent of every year but the 5th., interest is due; for the 4th. year, one year's interest, for the 3d, two years' interest, and so on, in Arithmetical Progression. The question falls, therefore, under cn. $24-the com. dif. 600+24×4=696 largest term. 696+600 X5÷2=83,240 Ans. Hence, to find the amount of an annuity in arrears, FIND THE SUM OF AN ARITHMETICAL PROGRESSION, OF WHICH THE ANNUITY IS THE FIRST TERM, THE NUMBER OF YEARS, THE NUMBER OF TERMS, AND THE ANNUAL INTEREST, THE RATIO; AND IT WILL BE THE AMOUNT REQUIRED. 2. Find the amount of an annuity of $50, for 7 yrs. at 5 per ct. Ans. $402.50. 3. What is the amount of an annuity of $250 for 7 yrs. at 6 per ct. payable half yearly? Ans. $2,091.25. NOTE. As the annuity is payable half yearly, it amounts to the same sum it would, if the years were twice as many, and the rate half as great. 4. If a rent of $200 remain unpaid 8 years, what will it amount to? Ans. $1,936. 5. A man sold a pension of $100 to continue 5 years, allowing the buyer 6 per ct. What did he receive? We may consider the annuity as five separate debts, due at the end of 1, 2, or 3 years, &c. Discounting at 6 per ct. on each, (§ LXXXII) we have 1st. year, present worth, $94.3396; 2d. year, $89.2857; 3d. $84.7457; 4th. $80.6451; 5th. $76.9230. The amount of these, 425.9391, is the amount of the annuity, required. Hence, to find the present worth of an annuity, for a given time, OF THE PRESENT WORTHS THUS FOUND WILL BE THE ANSWER. 6. Find the present worth of a pension of $500 for 4 years at 5 per cent. Ans. $1,782.185+ 7. Find the present worth of a salary of $200, to continue 3 yrs. at 4 per ct. Ans. $556.063, NOTE. The estimation of the present worths of Annuities at Simple Interest, may easily be shown to be unreasonable and unjust. For the price of an annuity of $100 found in this way, for 40 years, at 6 per ct., will amount to a sum, which, put at interest, would draw a greater annuity than the given one; and one which would last forever, instead of forty years. It is therefore most equitable to allow Compound Interest. ANNUITIES AT COMPOUND INTEREST. 1. An annuity of $100 was left 4 years unpaid. What was then due on it at 6 per ct. compound interest? In finding an amount at compound interest, we multiply continually by the rate per ct.+1, (§ LXXXVIII) till the number of multiplications is equal to the number of years. Now, in calculating an annuity, we have for the last year, the annuity without interest; for the next preceding, the amount of the annuity for 1 yr.; for the next preceding still, the amount for 2 yrs., and so on. Thus, at com. pound interest, an annuity in arrears forms a Geometrical Progression, whose ratio is the rate per ct.+1, and whose terms equal the years in number. In the above example then, 100×1.06-119.101. 119.101×1.06-100÷.06-$437.45 Ans. Hence, to find the amount of an annuity in arrears at compound interest, 3 FIND THE SUm of a geomeTRICAL SERIES, WHOSE FIRST TERM IS THE ANNUITY, AND WHOSE RATIO, THE RATE PER CENT.+1, FOR AS MANY TERMS AS THERE ARE YEARS. THIS WILL BE THE AMOUNT REQUIRED. 2. Find the amount of $150 annuity, for 4 years, at 10 per cent. A. $696.15 3. Find the amount of an annuity of $40, for 5 years, at 5 per A. $221.02525. cent. 4. Find the amount $50 annuity, for 7 years, at 4 per cent. A. $394.915 For convenience, has been calculated the following TABLE OF MULTIPLIERS, FOR FINDING THE AMOUNT OF AN ANNUITY, FOR ANY NUMBER OF YEARS, FROM 1 TO 40, at 6 per cent. Yrs.16 per cent. Yrs. 10 per cent. Yrs. 6 per cent. Yrs.16 per cent. 1 1.0000 11 14.9716 21 39.9927 31 84.8016 2 2.0600!! 12 16.8699 2243.3922|| 32 90.8897 33.1836 13 18.8821|| 23 46.9958 33 97.3431 4 4.3746 14 21.0150 2450.8155 34 104.1837 5 5.6371 15 23.2759 25 54.8645 35 111.4347 6 6.9753| 16|25.6725|| 26|59.1563|| 36 119.1200 7 8.3938 17 28.2123|| 27 63.7057|| 37127.2681 8 9.8974 1830.9056|| 28 68.5281|| 38135.9042 911.4913 1933.7599|| 29 73.6397 39 145.0584 10 13.1807 20/36.7855 30 79.0581 40 154.7619 cent. 5. Find the amount of an annuity of $150, for 3 years, at 6 per A. $477.54. 6. Find the amount of $500 annuity, for 24 years, at 6 per cent. A. $25,407.75. 7. Find the amount of $100 annuity, for 49 years, at 6 per cent. A. $26,172.08. NOTE. If the time given be greater than 40 yrs., calculate for 40, and consider this amount a debt at compound interest, for the remaining time, which calculate accordingly. Then calculate the amount for the remaining time, as though the annuity commenced again, and add this sum to the last. 8. A pension of $100 for 4 years, was sold, the buyer being allowed 6 per ct. compound interest, for his money. What did the seller receive ? It is evident that the present worth is a sum, which, at compound interest would, in 4 years, produce the amount of the given annuity, for the same time. This amount is $437.46. To find an amount at compound interest, we multiply a sum by the rate per cent. +1, as many times successively as there are years, or, in other words, we multiply by that power of the rate per cent.+1, whose index is the number of years. To find a present worth, we must, manifestly, reverse this process, and divide by the same power. Then $437.46÷1.26247 (=1.064) =$346.511 Ans. Hence, to find the present worth of an annuity, FIND THE AMOUNT IN ARREARS FOR THE GIVEN TIME, AND DIVIDE IT BY THAT POWER OF THE RATE PER CENT.+1, WHOSE INDEX IS THE NUMBER OF YEARS. NOTE. This power may be found in the table of multipliers for compound (SLXXXVIII.) interest. 9. Find the present worth of a $40 annuity, to continue 5 yrs. at 5 per cent. A. $173.173. 10. Find the present worth of $100 annuity, for 20 yrs. at 5 per A. $1,246.22. cent. For convenience, has been calculated the following TABLE OF MULTIPLIERS, FOR FINDING THE PRESENT WORTH OF AN ANNUITY FOR ANY NUMBER OF YEARS, FROM 1 TO 20, AT 5 AND 6 PER CENT. Years. 5 per cent. Years. 5 per cent. 6 per cent. 8.30641 7.88687 8.86325 8.38384 9.39357 8.85268 9.89864 9.29498 10.37966 9.71225 6 per cent. 0.95238 0.94339|| 11 2 1.85941 1.83339|| 12 3 2.72325 2.67301 13 4 3.54595 3.46510 14 5 4.32948 4.21236 15 5.07569 4.91732 16 10.83777 10.10589) 5.78637 5.58238| 17 11.27407|10.47726 6.46321 6.20979|| 18 |11.68958 10.82760 9 7.10782 6.80169 19 12.08532 11.15811 10 7.72173 7.36008|| 20 |12.46221 11.46992 11. Find the present worth of an annuity of $21.54, for 7 yrs. at 6 per cent. A. $120.244 12. Find the present worth of an annuity of $100, to continue 12 years, at 6 per cent. A. $838.384. 13. Find the present worth of an annuity of $936, for 20 yrs. at 5 per cent. A. $11,664.629 As any annuity multiplied by one of the numbers in the last table will give the present worth of that annuity, so it is evident that, any present worth, divided by the same number, will give the annuity itself. Hence, if I wish to discover of what annuity any given sum is the present worth, that is, what annuity any given sum will buy, I have only to use the above table, as a table of divisors, instead of multipliers. 14. What annuity, to continue 5 years, will $432,948 purchase, when money is worth 5 per cent. ? A. $100,000. 15. What annuity, to continue 19 years, will $6,694.866 purchase, when money will bring 6 per cent.? A. $600. When an annuity is to commence at some future time, it is said to be in reversion. An annuity in reversion, is evidently not worth as much as one of the same amount, which commences immediately. For if the present worth of the annuity be calculated as usual, it will be what the annuity is worth at the time it commences; and as that time is still future, we must discount for the intervening space, in order to obtain the true present worth. Hence, to find ihe pres ent worth of an annuity in reversion, FIND THE PRESENT WORTH AS USUAL, AND DISCOUNT UPON IT FOR THE TIME OF REVERSION, |