7. What is the sum of the descending series 3, 1, 1, 1, 27, &c., extended to infinity? It is evident the last term must become 0, or indefinitely near to nothing; therefore, the extremes are 3 and 0, and the ratio 3. Ans. 4. 8. What is the value of the infinite series 1+++ 4, &c. ? Ans. 14. 9. What is the value of the infinite series, To + TO + τοσο + τσόσο, &c., or, what is the same, the decimal '11111, &c., continually repeated? Ans.. 10. What is the value of the infinite series, Tοσ + τοσο, &c., descending by the ratio 100, or, which is the same, the repeating decimal '020202, &c. ? Ans. 11. A gentleman, whose daughter was married on a new year's day, gave her a dollar, promising to triple it on the first day of each month in the year; to how much did her portion amount? Here, before finding the amount of the series, we must find the last term, as directed in the rule after ex. 1. Ans. $2656720. The two processes of finding the last term, and the amount, may, however, be conveniently reduced to one, thus: When the first term, the ratio, and the number of terms, are given, to find the sum or amount of the series, Raise the ratio to a power whose index is equal to the number of terms, from which subtract 1; divide the remainder by the ratio, less 1, and the quotient, multiplied by the first term, will be the answer. Applying this rule to the last example, 312 = 531441, and X1265720. Ans. $2656720, as before. 531441-1 3-1 12. A man agrees to serve a farmer 40 years without any other reward than 1 kernel of corn for the first year, 10 for the second year, and so on, in 10 fold ratio, till the end of the time; what will be the amount of his wages, allowing 1000 kernels to a pint, and supposing he sells his corn at 50 cents per bushel? 1040-1 x1 = {1,111,111,111,111,111,111,111,111, 111,111,111,111,111 kernels. 10-1 Ans. $8,680,555,555,555,555,555,555,555,555,555,555 5551 13. A gentleman, dying, lest his estate to his 5 sons, to the youngest $1000, to the second $ 1500, and ordered, that each son should exceed the younger by the ratio of 11; what was the amount of the estate? Note. Before finding the power of the ratio 14, it may be reduced to an improper fraction = &, or to a decimal, 1'5. 25-1 -1 × 1000 = $131872; or, $13187'50, Answer. 1655-1 Compound Interest by Progression. × 1000 = ↑ 114. 1. What is the amount of $ 4, for 5 years, at 6 per cent. compound interest? We have seen, (192,) that compound interest is that, which arises from adding the interest to the principal at the close of each year, and, for the next year, casting the interest on that amount, and so on. The amount of $1 for 1 year is $1'06; if the principal, therefore, be multiplied by 1'06, the product will be its amount for 1 year; this amount, multiplied by 1'06, will give the amount (compound interest) for 2 years; and this second amount, multiplied by 1'06, will give the amount for 3 years; and so on. Hence, the several amounts, arising from any sum at compound interest, form a geometrical series, of which the principal is the first term; the amount of $1 or 1 £., &c., at the given rate per cent., is the ratio; the time, in years, is 1 less than the number of terms; and the last amount is the last term. The last question may be resolved into s:-If the first term be 4, the number of terms 6, and the ratio 1'06, what is the last term? 1'065=1'338, and 1'338×4=$56352+. Ans. $56352. Note 1. The powers of the amounts of $ 1, at 5 and at 6 per cent., may be taken from the table, under 11 91. Thus, opposite 5 years, under 6 per cent., you find 1'338, &c. Note 2. The several processes may be conveniently exhibited by the use of letters; thus: Let P. represent the Principal. ..... ..... .... R. the Ratio, or the amount of $ 1, &c. for 1 year. т. .......... the Time, in years. A. the Amount. When two or more letters are joined together, like a word, U they are to be multiplied together. Thus PR. implies, tha* the principal is to be multiplied by the ratio. When one letter is placed above another, like the index of a power, the first is to be raised to a power, whose index is denoted by the second. Thus RT. implies, that the ratio is to be raised to a power, whose index shall be equal to the time, that is, the number of years. 2. What is the amount of 40 dollars for 11 years, at 5 per cent. compound interest? RT × P. = A.; therefore, 1'0511 × 40 = 68'4. Ans. $68'40. 3. What is the amount of $6 for 4 years, at 10 per cent. Ans. $87841 compound interest ? 4. If the amount of a certain sum for 5 years, at 6 per cent. compound interest, be $5'352, what is that sum, or principal? If the number of terms be 6, the ratio 1'06, and the last term 5'352, what is the first term ? This question is the reverse of the last; therefore, A. 5'352 = P.; or, RT. 1'338 5. What principal, at 10 per cent. compound interest, will amount, in 4 years, to $8'7846? Ans. $6. 6. What is the present worth of $68'40, due 11 years hence, discounting at the rate of 5 per cent. compound interest? Ans. $40. 7. At what rate ne cent. will $6 amount to $ 8'7846 in 4 years? If the first term be 6, the last term 8'7846, and the number of terms 5, what is the ratio? A. P. = RT, that is, 8'7846 6 = 1'4641 = the 4th power of the ratio; and then, by extracting the 4th root, we obtain 1'10 for the ratio. Ans. 10 per cent. 8. In what time will $6 amount to $867846, at 10 per cent. compound interest? A. P. = RT, that is, 867846 6 = 14641 = 1'10T.; therefore, if we divide 1'4641 by 1'10, and then divide the quotient thence arising by 1'10, and so on, till we obtain a quotient that will not contain 1'10, the number of these divisions will be the number of years. Ans. 4 years. 9. At 5 per cent. compound interest, in what time will $40 amount to $68'40? Having found the power of the ratio 1'05, as before, which is 1'71, you may look for this number in the table, under the given rate, 5 per cent., and against it you will find the number of years. Ans. 11 years. 10. At 6 per cent. compound interest, in what time will $4 amount to $5'352? Ans. 5 years. Annuities at Compound Interest. 115. It may not be amiss, in this place, briefly to show the application of compound interest, in computing the amount and present worth of annuities. An ANNUITY is a sum payable at regular periods, of one year each, either for a certain number of years, or during the life of the pensioner, or forever. When annuities, rents, &c. are not paid at the time they become due, they are said to be in arrears. The sum of all the annuities, rents, &c. remaining un paid, together with the interest on each, for the time they have remained due, is called the amount. 1. What is the amount of an annual pension of $ 100 which has remained unpaid 4 years, allowing 6 per cent. compound interest? The last year's pension will be $ 100, without interest the last but one will be the amount of $100 for 1 year; the last but two the amount (compound interest) of $ 100 for 2 years, and so on; and the sum of these several amounts will be the answer. We have then a series of amounts, that is, a geometrical series, ( 114,) to find the sum of all the terms. If the first term be 100, the number of terms 4, and the ratio 1'06, what is the sum of all the terms ? Consult the rule, under T 113, ex. 11. 1'064-1 '06 × 100 = 437645. Ans. $437'45. Hence, when the annuity, the time, and rate per cent. are given, to find the amount,-RAISE the ratio (the amount of : $1, &c. for 1 year) to a power denoted by the number of years; from this power subtract 1; then divide the remainder by the ratio, less 1, and the quotient, multiplied by the annuity, will be the amount. Note. The powers of the amounts, at 5 and 6 per cent up to the 24th, may be taken from the table, under 11 91. 2. What is the amount of an annuity of $50, it being in arrears 20 years, allowing 5 per cent. compound interest? Ans. $ 1653′29 3. If the annual rent of a house, which is $150, be in arrears 4 years, what is the amount, allowing 10 per cent. compound interest ? Ans. $696'15. 4. To how much would a salary of $500 per annum amount in 14 years, the money being improved at 6 per cent. compound interest? years? - in 22 years? in 10 years? -in 20 in 24 years? Ans. to the last, $25407675. 116. If the annuity is paid in advance, or if it be bought at the beginning of the first year, the sum whicn ought to be given for it is called the present worth. 5. What is the present worth of an annual pension of $100, to continue 4 years, allowing 6 per cent. compound interest? The present worth is, evidently, a sum which, at 6 per cent. compound interest, would, in 4 years, produce an amount equal to the amount of the annuity in arrears the same time. By the last rule, we find the amount = $437'45, and by the directions under 1 114, ex. 4, we find the present worth = $346'51. Ans. $346'51. Hence, to find the present worth of any annuity,--First find its amount in arrears for the whole time; this amount, divided by that power of the ratio denoted by the number of years, will give the present worth. 6. What is the present worth of an annual salary of $ 100 o continue 20 years, allowing 5 per cent.? Ans. $1246 22. |