8. 9 XLI. GEOMETRICAL PROGRESSION. Art. 297. When there are three or more numbers, and the same quotient is obtained by dividing the second by the first, the third by the second, and the fourth by the third, &c., these numbers are in Geometrical Progression, and may be called a Geometrical Series. Thus, 2, 4, 16, 32, 64, 64, 32, 16, 8, 4, 2. The former is called an ascending series, and the latter a descending series. In the first series the quotient is 2, and is called the ratio ; in the second, it is 1. Hence, if the series is ascending, the quotient is more than unity; if it is descending, it is less than unity. The first and last terms of a series are called extremes, and the other terms, means. Any three of the five following things being given, the other two may be found : 1st. The first term, or first extreme; Art. 298. One of the extremes, the ratio, and the number of terms being given, to find the other extreme. ILLUSTRATION. Let the first term be 2, the ratio 3, and the number of terms 7. It is evident, that, if we multiply the first term by the ratio, the product will be the second term in the series; and, if we multiply the second term by the ratio, the product will be the third term; and, in this manner, we may carry the series to any desirable extent. By examining the following series, we find that 2, carried to the 7th term, is 1458; thus, QUESTIONS. — Art. 297. When are numbers in geometrical progression ? What is an ascending series? What a descending series? What is the ratio of a progression? Is the ratio greater or less than unity, in an ascending series? In a descending series? What are the extremes of a series ? What the means? What five things are mentioned, any three of which being given, the other two may be found 1 3 5 6 182 Ž & 18 54 486 1458 The factors of 1458, are 3, 3, 3, 3, 3, 3, and 2, the last of which is the first term of the series, and the others the ratio repeated a number of times 1 less than the number of terms. But multiplying these factors together is the same as raising the ratio to the sixth power, and then multiplying that power by the first term. Hence the following RULE. Raise the ratio to a power whose index is equal to the number of terms less one; then multiply this power by the first term, and the product is the last term, or other extreme. Note.— This rule may be applied in computing compound interest, the principal being the first term, the amount of one dollar for one year, the ratio, the time, in years, one less than the number of terms, and the amount the last term. EXAMPLES FOR PRACTICE. 1. The first term of a series is 1458, the number of terms 7, and the ratio ; what is the last term ? Ans. 2 OPERATION. Ratio (3)6 =qzgi x 1458=48=2, the last term. 2. If the first term of a series is 4, the ratio 5, and the number of terms 7, what is the last term ? Ans. 62500. 3. If the first term of a series is 28672, the ratio 4, and the number of terms 7, what is the last term ? Ans. 7. 4. The first term of a series is 5, the ratio 4, and the number of terms is 8; required the last time. Ans. 81920. 5. If the first term of a series is 10, the ratio 20, and the number of terms 5, what is the last term ? Ans. 1600000. 6. If the first term of a series is 30, the ratio 1.06, and the number of terms 6, what is the last term ? Ans. 40.146767328. 7. What is the amount of $ 1728 for 5 years, at 6 per cent., compound interest? Ans. $ 2312.453798+. 8. What is the amount of $328.90, for 4 years, at 5 per cent., compound interest ? Ans. $399.78+. 9. A gentleman purchased a lot of land containing 15 acres, agreeing to pay for the whole what the last acre would come QUESTIONS. — Art. 298. What is the rule for finding the other extreme, one of the extremes, the ratio, and number of terms being given? To what muy this rule be applied ? to, reckoning 5 cents for the first acre, 15 cents for the second, and so on, in a three-fold ratio. What did the lot cost him? Ans. $ 239148.45. Art. 299. To find the sum of all the terms, the first term, the ratio, and the number of terms being given. ILLUSTRATION. -- Let it be required to find the sum of the following series : 2, 6, 18, 54. If we multiply each term of this series by the ratio 3, the products will be 6, 18, 54, 162, forming a second series, whose sum is three times the sum of the first series; and the difference between these two series is twice the sum of the first series. Thus, 6, 18, 54, 162, the second series. 2, 6, 18, 54, the first series. 2, 0, 0, 0, 162--2=160, difference of the two series. Now, since this difference is twice the sum of the first series, one half this difference will be the sum of the first series; thus, 160 ; 2=80. It will be observed, by examining the operation above, that, if we had simply multiplied 54, the last term of the first series, by the ratio 3, and subtracted 2, the first term, from it, we should have obtained 160; and this being divided by the ratio, 3 less 1, would have given 80, the same number as before, for the sum of the first series. Hence the Rule. Find the last term as in the preceding article, multiply it by the ratio, and from the product subtract the first term. Then divide this remainder by the ratio less 1, and the quotient will be the sum of the series. Or, Rule II. 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 given extreme, will be the sum of the series required. NOTE 1. -If the ratio is less than a unit, the product of the last term multiplied by the ratio must be subtracted from the first term ; and to obtain the divisor, the ratio must be subtracted from unity, or 1. QUESTIONS. Art. 299. What is the rule for finding the sum of all the terms, the first terin, ratio, and number of terms being given? If the ratio is less than a unit, what musi be done with the product of the last term multiplied by the ratio ? How is the divisor obtained when the ratio is less than 1 ? NOTE 2. - If the second rule is employed, when the ratio is less than one, its power, denoted by the number of terms, must be subtracted from 1, and the remainder divided by the difference between 1 and the ratio. EXAMPLES FOR PRACTICE. 1. If the first term of a series is 12, the ratio 3, and the number of terms 8, what is the sum of the series? Ans. 39360. OPERATION. Ratio 37 X 12=26244, the last term; 26244 x3=78732; 78732 - 12=78720; 78720 • (3 — 1) = 39360, the sum of the series. 2. The first term of a series is 5, the ratio g, and the number of terms 6; required the sum of the series. Ans. 1314. OPERATION. Ratio (3)" x 5=14, the last term; 368 X = 25; 5. * ; * = (1 — ş) == 1311$, the sum of the series. 3. If the first term of a series is 8, the ratio 4, and the number of terms 7, required the sum of the series. Ans. 43688. 4. If the first term is 10, the ratio è, and the number of terms 5, what is the sum of the series? Ans. 30.05 5. If the first term is 18, the ratio 1.06, and the number of terms 4, what is the sum of the series? Ans. 78.743+. 6. When the first term is $ 144, the ratio $ 1.05, and the number of terms 5, what is the sum of the series? Ans. $ 795.6909. 7. D. Baldwin agreed to labor for E. Thayer for 6 months. For the first month he was to receive $3, and each succeeding month's wages were to be increased by f of his wages for the month next preceding; required the sum he received for his 6 months' labor. Ans. $9137 8. A lady, wishing to purchase 10 yards of silk for a new dress, thought $ 1.00 per yard too high a price; she, however, agreed to give 1 cent for the first yard, 4 for the second, 16 for the third, and so on, in a four-fold ratio; what was the cost of the dress? Ans. $ 3495.25. 25 ANNUITIES AT COMPOUND INTEREST BY GEOMETRICAL PROGRESSION. Art. 300. WHEN compound interest is reckoned on an annuity in arrears, the annuity is said to be at compound interest; and the amounts of the several payments form a geometrical series, of which the annuity is the first term, the amount of $ 1.00 for one year the ratio, the years the number of terms, and the amount of the annuity, the sum of the series. Hence, Art. 301. To find the amount of an annuity at compound interest, we have the following Rule I. Find the sum of the series by either of the preceding rules, (Art. 299.) Or, RULE II. Multiply the amount of $1.00, for the given time, found in the table, by the annuity, and the product will be the required amount. TABLE, Years. Years. 1 2 3 4 7 8 9 10 11 12 13 14 15 16 17 18 19 20 5 per cent. 1.000000 2.050000 3.152500 4.310125 5.525631 6.801913 8.142008 9.549109 11.026564 12.577893 14.206787 15.917127 17.712983 19.598632 21.578564 23.657492 25.840366 28.132385 30.539004 33.065954 6 per cent. 1.000000 2.060000 3.183600 4.374616 5.637093 6.975319 8.393838 9.897468 11.491316 13.180795 14.971643 16.869941 18.882138 21.015066 23.275970 25.672528 28.212880 30.905653 33.759992 36.785591 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 5 per cent. 35.719252 38.505214 41.430475 44.501999 47.727099 51.113454 54.669126 58.402583 62.322712 66.438847 70.760790 75.298829 80.063771 85.066959 90.220307 95.836323 101.628139 107.709546 114.095023 120.799774 6 per cent. 39.992727 43.392290 46.995828 50.815577 54.864512 59.156383 63.705766 68.528112 73.639798 79.058186 84.801677 90.889778 97.343165 104.183755 111.434780 119.120867 127.268119 135.904206 145.058158 154.761966 QUESTIONS. - Art. 300. When is an annuity said to be at compound inter est? What do the amounts of the several payments form? What is the first term of the series? What the ratio ? What the number of ternis ? What the sum of the series ?— Art. 301. What is the first rule for finding the amount of an annuity? What the second ? What does the table show ? |