CASE II. The first term, the last term, (or the extremes,) and the ratio given, to find the sum of the series. RULE. Multiply the last term by the ratio; from the product subtract the first term, and divide the remainder by the ratio, less 1, and the quotient will be the sum of all the terms. EXAMPLES. 1. A man bought 6 yards of cloth, giving 2 cents for the first yard, 6 cents for the second, and so on in triple proportion; what did the last yard cost, and how much was the cost of the whole? By Case I, the cost of the last yard is 486 cents. 2. If the first term of a geometricial series be 3, and the last term 6144, and the ratio 2, what is the sum of all the terms? Ans. 12285. 3. If the extremes of a geometrical series be 10, and 196830, and the ratio 3, what is the sum of all the terms? Ans. 295240. (Note. In the following examples, the scholar must find the last term of the series by Case I; then find the sum of all the terms, as above.) 4. A man purchased a valuable tract of land containing fifteen acres, agreeing to give 1 dollar for the first acre, 4 dollars for the second, and so on in a quadruple, or fourfold proportion; what did the whole tract cost him? Ans. $357913941. 5. What debt can be discharged in a year, by paying 1 cent the first month, 10 cents the second, and so on in a tenfold proportion? Ans. $1111111111,11. 6. If a pound (12oz.) of gold be sold at the rate of 2 cents for the first ounce, 8 cents for the second, 32 cents for the third, and so on in a fourfold proportion to the last; will it amount to? what Ans. $111848,10. 7. A merchant sold 14 yards of Italian silk at the rate of 4 cents for the first yard, 12 cents for the second, and so on in geometrical progression; how much did the last yard come to, and what did the whole amount to? Ans. The last come to $63772,92, and the whole $95659,36. 8. A man bought a horse, and by agreement was to give a cent for the first nail, two for the second, four for the third, and so on; there were four shoes, and eight nails in each shoe; what did the horse come to at that rate? Ans. $42949672,95. 9. A thresher worked 20 days for a farmer, and received for the first day's work 4 barley-corns, for the second 12 barley-corns, for the third 36 barley-corns, and so on in triple geometrical progression; what did his 20 day's labour come to, allowing 7680 barley-corns to make a pint, and the whole quantity to be sold at 50cts. per bushel ? Ans. $7093,50, rejecting remainders. 10. If a body put in motion move 1 inch the first second of time, 3 inches the second, 9 inches the 3d second of time, and thus continue to increase its motion in triple proportion, geometrical, how many yards will it move in the term of half a minute? Ans. 2859599056870yds. Oft. 4 inches, which is no less then one thousand six hundred and twenty-four millions of miles. Questions. 1. What is Geometrical Progression? 2. What is the multiplier or divisor in Geometrical Progression called? 3. How many parts are there in Geometrical Progression? 4. Having the first term, the ratio, and the number of terms given, how do you find the last term? 5. Having the first term, the last term, (or the extremes,) and the ratio given, how do you find the sum of the series? ANNUITIES. To find the amount of an annuity at Simple Interest, by Arithmetical Progression. RULE. Make 1 the first term of an arithmetical series, and the ratio the common difference-multiply the common difference by the number of terms less 1, and to the product add the first term for the last. Then find the sum of all the terms by the Rule, Case III. page 205, which will be the amount of $1 annuity for the given number of years. Multiply this amount by the given sum for the whole amount. CASE II. To find the present worth of annuities. The present worth here spoken of, is such a sum as if put at compound interest for the given rate and time will be equal to the amount of the given annuity. RULE. Raise the ratio to a power equal to the given number of years. Then divide the annuity by that power, and subtract the quotient from the annuity, and divide the remainder by the ratio, less 1. EXAMPLES. 1. What sum of ready money will purchase a yearly annuity of $60, to continue 4 years, at 6 per cent? Ratio 1,06 × 1,06 × 1,06 × 1,06= 1,26247696)60,00000000000(47,5256+ then from the annuity 60,0000 subtract this quotient 47,5256 ratio 1,06—1,06)12,4744 Ans. $207,906+$207,90c. 6m.+* TABLE II. Showing the present worth of an annuity of $1 or £1, at 5 and 6 per cent for any number of years from 1 to 32. yrs. 5 per cent. 6 per cent. 6 per cent. 0,943396 17 11,274066 10,477260 1,833393 18 11,689587|10,827603 2,673012 19 12,085321 11,158116 3,465106 20 12,462210 11,469921 4,21236421 12,82115311,764077 4,917324 22 13,16300312,041582 5,582381 23 13,48857412,303380 6,209794 24 13,798642 12,550357 6,801692 25 14,093944 12,783356 7,36008726 14,375185 13,003166 7,886875 27 14,643034 | 13,210534 8,383844 28 14,898127 | 13,406164 8,85268329 15,141073 13,590721 9,294984 30 15,372451 13,764831 9,712249 31 15,592810 13,929080 10,10589532 15,80268114,084042 RULE. Multiply the present worth, found in Table II. opposite the given year, and under the rate per cent, by the given annuity. 2. What is the present worth of an annuity of $200, to continue 10 years, at 6 per cent ? by Table II. present worth of $1 for 10yrs. 7,360087 Ans. $1472,1c.74m. X 200 $1472,017400 3. How much must be paid in ready money, to purchase an annuity of $156 to continue 8 years at 5 per cent ? Ans. 1008,26c.1m.+ 4. What is the present worth of an annual pension of $96 a year, to continue 5 years at 6 per cent compound interest? Ans. $404,38c.6m.+. 5. What ready money will purchase an annuity of $112 to continue 30 years, at 5 per cent compound interest? Ans. $1721,71,4+. CASE III. To find the present worth of annuities, leases, &c. taken in reversion at Compound Interest. Observation.- Annuities in reversion, are those which do not commence till some particular event has happened, or until the expiration of a certain time. RULE. 1. Divide the annuity by the power of the ratio equal to the time of its continuance, and subtract this quotient from the annuity, and divide the remainder by the ratio less 1, and the quotient will be the present worth to commence immediately. 2. Divide the quotient by the power of the ratio equal to the time of reversion, (or the time to come before the annuity commences,) and the quotient will be the present worth of the annuity in reversion. 1. What is the present worth of $200, payable yearly for 4 years but not to commence, (that is being in reversion,) till the end of 2 years, at 6 per cent ? 4th power of 1,06 1 26247696)200,00000000(153,418732 153,418732 Ratio 1,06 less 1, ,06)41,581268 $ cts.m.. 2d power of 1,06 1,1236)693,0211(616,78,6+ It will be much easier to find the present worth of annuities in reversion, by 2d Table. Thus, find the present worth of $1, or £1, annuity for the sum of the time of reversion and the time of continuance added together; and from this present worth, subtract the present worth of $1 or £1 for the time of reversion, and multiply the remainder by the given annuity. 2. What is the present worth of $50 yearly annuity to continue 8 years, and to be in reversion 3 years at 5 per cent? Time of reversion, or the time before the annuity commences 3 years. Time of continuance 8 years. The present worth of $1 for 11 yrs. by 2d Table, 8,306414 3 yrs. reversion 2,723248 5,583166 X 50 Ans. $279,158300 3. What is the present worth of the reversion of a lease of $125 to continue 20 years, but not to commence till the end of 9 years, allowing 6 per cent? Ans. $848,62c.8+m. 4. What sum of ready money will purchase the reversion of an annuity of $60 to continue 15 years, but not to commence till the end of 10 years, at 5 per cent? CASE IV. Ans. $382,33c.2+m.. To find the present worth of freehold estates, or annuities to continue forever. RULE. As the rate per cent is to 100, so is the yearly rent or annuity to the value required. Or, divide the yearly rent |