The cube of the tens, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. As 91125 consists of five figures its cube root has two figures (Art. 446), and the cube of the tens of the root must be in the 91 (thousands). The greatest cube in 91 (thousands) is 64 (thousands), therefore the tens figure of the root must be 4. Now as the number 91125 equals the cube of the tens of the root, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units, if we subtract from 91125 the cube of the tens of the root, that is 403, or 64000, the remainder, 27125, must equal three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. In order, therefore, to find the units figure of the root we divide 27125 by three times the product of the square of the tens, that is, by 4800 as a trial divisor, obtaining 5 as the (probable) units figure of the root. As 3 (40a × 5) + 3 (40 × 52) + 58 = {3 (402) + 3 (40 × 5) +52 × 5, if to the trial divisor, 3 × 402, we add 3 (40 × 5) + 52, we shall obtain the true divisor, 5425, and if 5 times the true divisor equals 27125, the 5 is the units figure of the root. Similar reasoning applies, however many figures there are in the root. PROGRESSION. 581. A progression is a series of numbers which increase or decrease according to a fixed law. 582. The terms of a series are the several numbers that form the series. The first and last terms are called the extremes, and the others the means. ARITHMETICAL PROGRESSION. 583. Arithmetical Progression is a series of numbers increasing or decreasing by a common difference. Thus, and 2, 5, 8, 11, 14, 17, is an ascending series, 35, 30, 25, 20, 15, 10, is a descending series. 584. In Arithmetical Progression there are five elements of which any three being given, the other two can be found: 1. The first term. 2. The last term. 3. The common difference. 4. The number of terms. 5. The sum of all the terms. 585. In an ascending series, let 6 be the first term and 5 the common difference; 6 + 5 + 5 + 5 = 6 + 3 × 5 = 21, 4th term. Thus we see that, in an ascending series, the second term is found by adding the common difference once to the first term; the third term, by adding the common difference twice to the first term, etc. A similar explanation may be given when the series is descending. Hence, 586. The first term, common difference, and number of terms given, to find the last term. Rule. Multiply the common difference by the number of terms less one; add the product to the first term if the series is ascending, or subtract the product from the first term if the series is descending. 88. If the first term of an ascending series is 4, the common difference 3, and the number of terms 8, what is the last term? Ans. 25. 89. The first term of a descending series is 98, and the common difference 6; what is the 4th term? 90. What is the amount of $200, at 5 %, simple interest, for 18 years? 587. The extremes and number of terms given to find the common difference. By inspecting the formation of the series in Art. 585, it will be seen that the difference between the extremes is equal to the common difference multiplied by one less than the number of terms. Thus the difference between the 1st and 4th terms (21 6 = 15) is the sum of 3 equal additions; hence this difference divided by 3 (15 ÷ 3 = 5) gives one of these additions, that is the common difference. Hence, Rule. Divide the difference of the extremes by the number of terms less one. 91. The extremes of an arithmetical series are 4 and 55, the number of terms is 18; what is the common difference? and Ans. 3. 92. A man has 7 children whose ages form an arithmetical series; the youngest is 3 years old and the oldest 21; what is the difference of their ages ? 93. The amount of $300 at simple interest for 10 years is $450; what is the rate per cent? 588. The extremes and common difference given to find the number of terms. By Art. 585 it is evident that the difference of the extremes is the common difference multiplied by one less than the number of terms. Hence, conversely, Rule. Divide the difference of the extremes by the common difference, and add one to the quotient. 94. The extremes of an arithmetical series are 15 and 57, and the common difference is 6; what is the number of terms? Ans. 8. 95. The common difference in the ages of the children in a family is 2 years, the youngest is 5 years old, and the oldest 23; how many children in the family? 589. The extremes and number of terms given to find the sum of the series. Find one half of the product of the sum of the extremes multiplied by the number of terms. 96. The extremes of a series are 5 and 61, and the number of terms is 15; what is the sum of the series? Ans. 495. 97. How many strokes does a clock strike in 12 hours? GEOMETRICAL PROGRESSION. 590. Geometrical Progression is a series of numbers increasing or decreasing by a common ratio. Thus, 2, 6, 18, 54, 162, is an ascending series, 64, 32, 16, 8, 4, is a descending series. In the first series 3 is the ratio, and in the second. 591. In Geometrical Progression there are five elements, of which any three being given the other two can be found: 1. The first term. 2. The last term. 3. The ratio. 4. The number of terms. 5. The sum of all the terms. 592. The first term, ratio, and number of terms given, to find the last term. In a series, let 2 be the first term and 4 the ratio; In this series we see that the second term is found by multiplying the first term by the ratio; the third term, by multiplying the first by the square of the ratio; the fourth, by multiplying the first by the cube of the ratio, the index of the power of the ratio always being one less than the number of the term sought. A similar explanation may be given when the series is descending. Hence, Rule. Multiply the first term by that power of the ratio whose index is equal to the number of terms less one. |