CHAPTER XXIV. PROGRESSIONS. ARITHMETICAL PROGRESSION. 430. A series of numbers that increase or decrease by a common difference is called an Arithmetical Progression. Thus, the numbers 5, 8, 11, 14 are an arithmetical progression, since they increase by a common difference, 3; and the numbers 12, 10, 8, 6 are an arithmetical progression, since they decrease by a common difference, 2. The several numbers of a series are called its terms. 431. In the series 2, 5, 8, 11, 14, 17, 20 it is obvious that the second term, 5, is 2+(3 × 1); the third term, 8, is 2+(3×2); the fourth term, 11, is 2+ (3 × 3); the fifth term, 14, is 2+ (3 × 4); and so on. Hence, Any term may be found by multiplying the common difference by 1 less than the number of the term, and adding the product to the first term. = Thus, the twelfth term of the series 2, 5, 8..... will be 2 + (3 × 11) 35. In like manner, any term of a decreasing series will be found by subtracting this multiple of the difference from the first term. Thus, the eleventh term of the series 50, 46, 42..... will be 50 — (10 × 4) = 10. 1. The seventh term of the series 3, 5, 7..... 7. When the first term of a series is 5, and the common difference 21, find the thirteenth and eighteenth terms. Ex. When the fourth term of a series is 14, and the twelfth term 38, find the common difference. The difference between the fourth and twelfth terms will evidently be eight times the common difference. common difference will be 3814 = 3. Find the common difference in a series: Hence, the 8. Whose fourth term is 12 and seventh term 27. be 432. The sum of seven terms of the series 3, 5, 7..... will 3+ 5+ 7+9+11+13+15, or written in 15+13+11+ 9+ 7+ 5+ 3 reverse order, twice the sum Hence, the sum Therefore, 18+18+18+18+18+18+18=18x7. 2 433. It will be seen that 18 is the sum of 3 and 15; that is, the sum of the first and last terms; and that 7 is the number of terms. Hence, The sum of an arithmetical series may be found by multiplying one-half the sum of the first and last terms by the number of terms. Thus, the sum of eight terms of the series whose first term is 3, and last term 38, is to seven terms. 3. 8+7+7 3 + ··· to sixteen terms. ..... 4. 20+18+16} + ··· 5. The first twenty natural numbers. 6. The natural numbers from 37 to 53 inclusive. 7. A series of thirty terms, of which the first is 21 and the last 59. 8. The series whose first two terms are 3 and 9 and last 75. 9. A series of twenty terms whose third and fifth terms are 10 and 15. 10. A stone, when dropped from a height, falls through 16.1 ft. in the first second, 48.3 in the next, 80.5 in the third, and so on, in arithmetical progression. How far will it fall in the seventh second? and how far in 7 sec.? 11. A, who travels 8 mi. the first day, 11 the second, 14 the third, and so on, overtakes in 17 dys. B, who started at the same time, and travelled uniformly. What is B's rate per day? 12. One hundred stones lie in a straight line, 1 yd. apart. A boy starts at the first stone, brings each of the others in separately, and piles them with the first stone. How far does he travel? 434. If a represent the first term, d the common difference, the last term, n the number of terms, and s the sum of the terms, when any three of the five quantities are given, the other two may be found by means of the following expressions, called formulas: Ex. If the common difference is, the first term 13, and the last term 41, how many terms are there? Here d, a, and l are given, and n is required. If for d, l, and a in the first formula, 2, 41, and 13, respectively, be substituted, the result is By subtracting 13 from each side, 41 = 13 + (n − 1) × §. - 28 = (n − 1) × §. n = = 57. GEOMETRICAL PROGRESSION. 435. A series in which each term is obtained from the preceding term by multiplying it by a constant multiplier is called a Geometrical Progression. Thus, the numbers 3, 6, 12, 24..... are a geometrical progression, since each term is twice the preceding term; and the numbers 9, 3, 1, ..... are a geometrical progression, since each term is one-third of the preceding term. The constant multiplier is called the ratio of the progression. 436. In the series 3, 6, 12, 24, 48, 96, ..... it is obvious that the second term, 6, is 3 x 2; the third term, 12, is 3 x 22; the fourth term, 24, is 3 x 23; the fifth term, 48, is 3 x 24; and so on. Hence, Any term of a geometrical series may be found by multiplying the first term by that power of the ratio that is 1 less than the number of the term. Thus, the eighth term of the series 3, 6, 12..... will be 437. When any two consecutive terms of a series are given, the ratio is found by dividing the second term by the first. When two terms which are not consecutive are given, the ratio may be found as in the following example: Ex. Find the ratio in a series whose second and sixth terms are 5 and 80. 1. The eighth term of the series 2, 6, 18..... 6. Write the first three terms of the series whose fifth and sixth terms are 112 and 224, respectively. 7. The seventh and ninth terms of a series are 100 and 144, respectively. Find the twelfth term. 8. A capital of $1000 is increased by of itself each year. What will it be at the beginning of the fifth year? 9. A capital of $1000 is increased by 18 of itself each year. 6 What will it be at the beginning of the sixth year? |