a n 8 = Geometrical Progression. Arithmetical Progression. l= 1 1 (a + l)n P= (a l)" 2 Thus in comparing the first table with the second, we see that multiplication corresponds to addition ; division “ subtraction ; involution 66 multiplication ; evolution 66 division. If, therefore, we had a series of numbers bearing the same ratio to the natural series, as an Arithmetical to a Geome. trical Progression, the labor of multiplication would be reduced to that of simple addition, and involution to simple multiplication. Such a series constitutes a TABLE OF LOGA RITHMS. CHAPTER XIX. HARMONICAL PROGRESSION.* When three numbers are such that the first is to the third, as the difference of the first and second is to the difference of the second and third, they are said to be in HARMONICAL Proportion, and a series of numbers in continued harmonical proportion, constitutes a HARMONICAL PROGRESSION. The reciprocal of a number, is the quotient of 1 by the number. Thus } is the reciprocal of 2; 4 is the reciprocal of i; is the reciprocal of Ź, &c. The reciprocals of any equidifferent series form a harmonical proportion. PROBLEM I. Two numbers being given to find a third in harmonical proportion. * So called, because if a musical string be divided in harmonical proportion, the different parts will vibrate in unison. RULE. Consider the reciprocals of the numbers as two terms of an equidifferent series. The third term will be the reci. procal of the number sought. Find a third harmonical proportional to 120 and 40. The reciprocals are īżo, and 46 or 7io. The third term of the equidiffereat series is 12 oy and its reciprocal 24 is the harmonical proportional sought. 1. The first two terms of a harmonical progression are 60 and 30. Required the ten succeeding terms. 2. The first two terms of a harmonical proportion are 348075 and 69615. Find the six succeeding terms. PROBLEM II. To insert any number of harmonical means between two numbers. RULE. Find as many arithmetical means between the reciprocals of the given numbers. These means will be the reci. procals of the harmonical means. Insert 4 harmonical means between 20 and 120. The reciprocals are 24 and ily, or 127 and iżo. The four arithmetical means are jiji hili io and i žo, whose reciprocals are 24, 30, 40 and 60,—the desired harmonical means. 3. Insert 7 harmonical means between 630 and 5040. CHAPTER XX. ANNUITIES. Any sum of money to be paid regularly, at stated periods, is called an ANNUITY. The payment may be stipulated for a given number of years, in which case it is ca!led an an. nuity certain, or it may be dependent upon some particular circumstance, as the life of one or more individuals. The latter is called a contingent annuity. A perpetual annuity, is one which can only be terminated by the grantor, on the payment of a sum whose interest will be equivalent to the annuity. Of this character is the consolidated debt of England. An annuity in possession, is one on which there is a present claim; an annuity in reversion, or deferred annuity, is one that does not commence until the lapse of a stated time, or the occurrence of some uncertain event, as the death of an individual. The present worth of an annuity, is the sum which, at compound interest for the time of its duration, would amount to the sum of all the payments, each being placed at compound interest as it became due. PROBLEM I. To find the amount due on an annuity which has remained unpaid a given time. As the payments are all at compound interest, this case evidently falls under Problem II., in Geometrical Progression. RULE. Find the sum of a Geometrical Progression, in which the first term is the annuity, the ratio is the amount of $1.00 for the time that should elapse from one payment to another, and the number of terms is the number of pay. ments due. 1. If a person saves $250 per annum, and invests it at 7 per cent. compound interest, how much will he be worth at the end of 25 years? 2. What is the amount of an annuity of $500 payable semi-annually, forborn for 7 years, at 6 per cent. per annum? 3. What is the amount of a quarterly annuity of $400, in arrears for 5 per cent. ? 4. What is the amount of an annual salary of $1000 for 10 years, at 6 per cent. ? years, at 5 5. What is the amount of a quarterly rent of $200 for 3 years, at 6 per cent. ? 6. What is the amount of a pension of $400 a year, payable semi-annually, for 3 years and 6 months, at 7 per cent. per annum ? 7. What is the amount of a rent of $1500, ten years forborn, at 5 per cent. per year? 8. An estate that yields an annual income of $2000, is offered for sale for the amount of 10 years income at 6 per cent. compound interest. What is the price of the estate ? 9. What is the amount of a salary of $1300 for 16 years, at 5 per cent. compound interest ? PROBLEM II. To find the present worth of an annuity certain. What is the present worth of a semi-annual rent of $500, for 6 years at 6 per cent ? By the preceding Problem, we find that the rent at the expiration of the 6 years, will have amounted to $7096.015. The question is, therefore, to find the present worth of $7096.015 due in 6 years, at a compound interest of 3 per cent. semi-annually, which is done by Problem I., in Geometrical Progression. RULE. Divide the amount of the annuity, (found by Problem 1.) by the amount of $1.00 for the same time. 10. What is the present worth of an annual salary of $800 to continue 8 years, at 5 per cent. compound interest ? 11. What is the present worth of an annual income of $500, to continue 11 years, at 6 per cent. compound interest? 12. What is the present worth of a quarterly rent of $200, to continue 5 years, at 7 per cent, compound interest? 13. What is the present worth of a semi-annual pension of $175, to continue 9 years, at 41 per cent. compound interest? 14. What is the present worth of an annuity of $3000 for 7 years, at 3 per cent. compound interest ? 15. What sum invested at 6 per cent, compound interest, will yield me an income of $1600 per annum, for 25 years ? 16. What sum at 5 per cent, will yield an annual income of $1200 for 15 years ? 17. A gentleman wishes to present his estate to his children, reserving enough to yield $700 per annum for 15 years. How much must he reserve, allowing 5 per cent. compound interest ? PROBLEM III. The present worth of a perpetual annuity, is sum which would yield an interest equivalent to the annuity. As the interest is found by multiplying the principal by the rate, the principal may be found by dividing the interest by the rate. RULE. Divide the annuity by the rate per cent. 18. For how much should an estate that rents for $175 per year, be sold, to allow the purchaser 6 per cent. interest on his investment? 19. What is the par value of an annual income of £500 in the 4 per cent. consols.* ? 20. What sum of money must be laid out in the 3 per cent. consols., at 68 per cent, of the par value, to yield an income of £1000. 21. What sum of money at 43 per cent. will yield an annual interest of $620 ? 22. What sum, invested in an estate that rents for $400 per annum will yield an interest of 8 per cent. ? 23. A farm rents for $750 per annum. For what price should it be sold, when money is worth 6 per cent. a year? 24. What sum will build a wall worth $1000, and renew it every 15 years, at 5 per cent. compound interest? * Consols., is an abbreviation for the consolidated annuities of the British National Debt. |