§ 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, 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. 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; 2d. The last term, or last extreme; 4th. The ratio; 5th. The sum of the terms, or series. 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? 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 Ratio (3)675; = OPERATION. 5×1458-44582, 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, and the number of terms 7, what is the last term? 4. The first term of a series is 5, the ratio ber of terms is 8; required the last time. Ans. 7. 4, and the num 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 may 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, 2, 6, 18, 54, 2, 0, 0, 0, 162, the second series. the first series. 162-2160, 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, 1602 = 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 term, ratio, and number of terms being given? If the ratio is less than a unit, what must 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 × 12=26244, the last term; 26244 × 3=78732; 78732 1278720; 78720 ÷ (3 — 1)=39360, the sum of the series. 2. The first term of a series is 5, the ratio, and the number of terms 6; required the sum of the series. 160 OPERATION. 243 Ratio (3)5×59, the last term; 5-428-3325; 3325 ÷ (1-3)= 3322 = sum of the series. Ans. 1316. x=728; = 131, the 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. 30 65 Ans. 43688. 4. If the first term is 10, the ratio, and the number of terms 5, what is the sum of the series? 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 of his wages for the month next preceding; required the sum he received for his 6 months' labor. Ans. $9117. 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, Showing the amount of $1 annuity from 1 year to 40. 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? |