SECTION XVIII PROGRESSION. ART. 596. When there is a series of numbers such, that the ratios of the first to the second, of the second to the third, &c., are all equal, the numbers are said to be in Continued Proportion, or Progression. Progression is commonly divided into arithmetical and geometrical. Note. The terms arithmetical and geometrical are used simply to distinguish the different kinds of progression. They both belong equally to arithmetic and geometry. ARITHMETICAL PROGRESSION. 597. Numbers which increase or decrease by a common difference, are in arithmetical progression. (Art. 474. Obs.) OBS. 1. Arithmetical progression is sometimes called progression by difference, or equidifferent series. 2. When the numbers increase, the series is called ascending; as, 3, 5, 7, 9, 11, &c. When they decrease, the series is called descending; as, 11, 9, 7, 5, &c. 598. When four numbers are in arithmetical progression the sum of the extremes is equal to the sum of the means. Thus, if 5—3—9—7, then will 5+7=3+9. Again, if three numbers are in arithmetical progression, the sum of the extremes is double the mean. Thus, if 9-6-6-3, then will 9+3=6+6. 599. In any arithmetical progression, the sum of the two extremes is equal to the sum of any other two terms equally distant from the extremes, or equal to double the middle term, when the number of terms is odd. Thus, in the series 1, 3, 5, 7, 9, it is obvious that 1+9=3+7=5+5. 600. In an ascending series, each succeeding term is found by adding the common difference to the preceding term. Thus, if the first term is 3, and the common difference 2, the series is 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, &c. In a descending series, each succeeding term is found by subtracting the common difference from the preceding term. Thus, if the first term is 15, and the common difference 2, the series is 15, 13, 11, 9, 7, &c. 601. In arithmetical progression there are five parts to be considered, viz: the first term, the last term, the number of terms, the common difference, and the sum of all the terms. These parts have such a relation to each other, that if any three of them are given, the other two may be easily found. 602. If the sum of the two extremes of an arithmetical progression is multiplied by the number of the terms, the product will be double the sum of all the terms in the series. Take the series The same inverted The sums of the terms are 2, 4, 6, 8, 10, 12. 12, 10, 8, 6, 4, 2. 14, 14, 14, 14, 14, 14. Thus, the sum of all the terms in the double series, is equal to the sum of the extremes repeated as many times as there are terms; that is, the sum of the double series is equal to 12+2 multiplied by 6. But this is twice the sum of the single series. Hence, 603. To find the sum of all the terms, when the extremes and the number of terms are given. Multiply half the sum of the extremes by the number of terms, and the product will be the sum of the given series. OBS. The reason of this process is manifest from the preceding illustration. Ex. 1. The extremes of a series are 3 and 25, and the number of terms is 12: what is the sum of all the terms? Ans. 168. 2. What is the sum of the natural series of numbers, 1, 2, 3, 4, 5, &c., up to 100? 3. How many strokes does a common clock strike in 12 hours? 604. To find the common difference, when the extremes and the number of terms are given. Divide the difference of the extremes by the number of terms less 1, and the quotient will be the common difference required. OBS. The truth of this rule is manifest from Art. 602. 4. The extremes are 5 and 56, and the number of terms is 18: what is the common difference? Ans. 3. 5. If the extremes are 3 and 300, and the number of terms 10, what is the common difference? 605. To find the number of terms, when the extremes and common difference are given. Divide the difference of the extremes by the common difference, and the quotient increased by 1 will be the number of terms. OBS. The truth of this principle is manifest from the manner in which the successive terms of a series are formed. (Art. 600.) 6. If the extremes are 6 and 470, and the common difference is 8, what is the number of terms? Ans. 59. 7. If the extremes are 500 and 70, and the common difference is 10, what is the number of terms? 606. When the sum of the series, the number of terms, and one of the extremes are given, to find the other extreme. Divide twice the sum of the series by the number of terms, and from the quotient take the given extreme, OBS. The reason of this rule is manifest from Art. 602. 8. If the sum of a series is 576, the number of terms 24, and the first term 1, what is the last term? Ans. 47. 9. If the sum of a series is 1275, the number of terms 50, and the greater extreme 474, what is the less extreme? 607. To find any given term, when the first term and the common difference are given. Multiply the common difference by one less than the number of terms required; then if the series be ascending, add the product to the first term; but if it be descending, subtract it. OBS. The reason of this rule may be seen from the manner in which the succeeding terms of a series are formed. (Art. 600.) 10. If the first term of an ascending series is 7, and the common difference 3, what is the 41st term? Ans. 127. 11. If the first term of a descending series is 100, and the common difference 14, what is the 54th term? 12. If the first term of an ascending series is 7, and the com、 mon difference 5, what is the 100th term? 608. To find any given number of arithmetical means, when the extremes are given. Subtract the less extreme from the greater, and divide the remainder by 1 more than the number of means required; the quotient will be the common difference, which being continually added to the less extreme, or subtracted from the greater extreme, will give the mean terms required. One mean term may be found by taking half the sum of the extremes. (Art. 598.) OBS. This rule depends upon the same principle as that in Art. 604. 13. Required 3 arithmetical means between 7 and 35. 14. Required 6 arithmetical means between 1 and 99. GEOMETRICAL PROGRESSION. 609. Numbers which increase by a common multiplier, or decrease by a common divisor, are in Geometrical Progression. The numbers 4, 8, 16, 32, 64, &c., are in geometrical progression; and if each preceding term is multiplied by 2, the product will be the succeeding term; thus, 4×2=8; 8×2=16, &c. Again, if the order of this series be inverted, the proportion will still be preserved and the common multiplier become a common divisor. Thus, in the series 64, 32, 16, 8, &c., 64÷2=32; 32-2=16, &c. Note. If the first term and ratio are the same, the progression is simply a series of powers; as 2; 2×2; 2×2×2; 2×2×2×2, &c. OBS. 1. Geometrical Progression is geometrical proportion continued. It is therefore sometimes called continual proportionals, or progression by quotients. If the series increases it is called ascending; if it decreases, descending. 2. The numbers which form the series, are called the terms of the progression. The common multiplier, or divisor, is called the ratio. For most purposes, however, it will be more simple to consider the ratio as always a multiplier, either integral or fractional. Thus, in the series 64, 32, 16, &c., the ratio is either 2 considered as a divisor, or considered as a multiplier. 3. In Geometrical as well as in Arithmetical progression, there are five parts to be considered, viz: the first term, the last term, the number of terms, the ratio, and the sum of all the terms. These parts have such a relation to each other, that if any three of them are given, the other two may be easily found. 610. To find the last term, when the first term, the ratio, and the number of terms are given. Multiply the first term into that power of the ratio whose index is 1 less than the number of terms, and the product will be the last term required. OBS. 1. The reason of this process may be seen by adverting to the manner in which each successive term is formed. (Art. 609.) Thus, in the series 4, 8, 16, 32, &c., the 2d term 8=4X2; 16=4×2×2, or 4×22; 32=4×23, &c. 2. It will be seen that the several amounts in compound interest, form a geometrical series of which the principal is the 1st term; the amount of $1 for 1 year the ratio; and the number of years+1 the number of terms. Hence the required amount of compound interest may be found in the same way as the last term of a geometrical series. 1. If the first term of a geometrical progression is 2, and the ratio 4, what is the 5th term? 2. The first term is 64, and the ratio : Ans. 512. what is the 5th term? what is the 8th term? what is the 10th term? 4. The first term is 7, and the ratio 5: 5. A farmer hired a man for a year, agreeing to give him $1 for the 1st month, $2 for the 2d, $4 for the 3d, and so on, doubling his wages each month: how much did he give the last month? 6. What is the amount of $250, at 6 per cent., for 5 years compound int.? Of $500, at 7 per ct., for 6 years? Of $1000, at 5 per ct., for 10 years? 611. To find the sum of the series, when the ratio and the extremes are given. Multiply the greatest term into the ratio, from the product subtract the least term, and divide the remainder by the ratio less 1. OBS. 1. When the first term, the ratio, and the number of terms are given, to find the sum of the series we must first find the last term, then proceed as above. 2. The sum of an infinite series whose terms decrease by a common divisor, may be found by multiplying the greatest term into the ratio, and dividing the · product by the ratio less 1. The least term being infinitely small, is of no comparative value, and is therefore neglected. 7. What is the sum of the series, whose extremes are 5 and 1215, and the ratio 3? Ans. 1820. 8. The extremes of a series are 1 and 512, and the ratio 2 what is the sum of the series? |