3. Extract the cube root of 12 to 8 places of figures. 12(2.2894284 Def. 1. Numbers are said to form an Arithmetic Progression or series, or to be in Arithmetic Proportion, when the differences between any two consecutive terms of the series are the same. This difference is called the common difference. Def. 2. The first and last terms of a series are called the extremes, the other terms the means. 1. TO FIND ANY REQUIRED TERM OF AN ARITHMETIC SERIES, OF WHICH THE FIRST TERM AND THE COMMON DIFFERENCE ARE KNOWN. Rule. To the first term add the product of the common difference multiplied by a number less by one than the number of the term. EXAMPLE. Find the 9th term of the series, of which the 1st term is 13, and common difference is 2. 9th term = 13+2 X8= 13 + 16 = 29. ll. TO FIND THE SUM OF ANY NUMBER OF TERMS OF A GIVEN ARITH METIC SERIES. Rule. Find the last term, and multiply half the sum of the first and last terms by the number of terms. EXAMPLE Here first term=1: com. diff. = 2. 1 + 39 x 20 = 400. 2 .. sum lll. TO FIND ANY NUMBER OF ARITHMETIC MEANS BETWEEN TWO NUMBERS. Rule. Divide the difference between the last and first term by a number greater by one than the required number of means. The quotient is the common difference, which being known, the terms of the series are readily found. EXAMPLE. Insert 7 arithmetic means between 7 and 27. 27 — 7 20 1 Com. diff. = = 2– 7 +1 8 2 Def. 1. Numbers are said to form a Geometric Progression or Series, or to be in Geometric Proportion, when the ratios of any one term to the preceding are the same. This ratio is called the Common Ratio. Def. 2. The limit of the sum of any number of terms of an infinite Geometric Series, is the number to which the sum continually approaches, as the number of terms is increased, but which the sum of any number of terms whatever can never equal or exceed. The common ratio here is always a proper fraction. 1. TO FIND ANY REQUIRED TERM OF A GIVEN GEOMETRIC PROGRESSION. Rule. Raise the common ratio to a power one less than the number of the required term, and multiply by the first term. EXAMPLE. Here Ist term = 2: common ratio = : 2, = 16 X 8= 128. II. TO FIND THE SUM OF ANY NUMBER OF TERMS OF A GEOMETRIC SERIES. Rule. Divide the difference between the first term, and the last term multiplied by the common ratio, by the difference between 1 and the common ratio. EXAMPLES. 1. Find the sum of 8 terms of the series 1, 3, 9, &c. Last term=1 X 37 = 2187. 2187 - 1 2186 ... Sum = 1093. 1 2. Find the sum of 9 terms of the series 1, ], }, &c. 1 1 Last term=1 X 28 256 256 1 255 127 .. Sum =1256 128 128 128 1-256 III. TO FIND THE LIMIT OF THE SUM OF AN INFINITE GEOMETRIC SERIES. Rule. Divide the first term by the difference between 1 and the common ratio, EXAMPLE. 1 3 IV. TO FIND ANY NUMBER OF GEOMETRIC MEANS BETWEEN TWO NUMBERS. Rule. Divide the last by the first term ; extract a root of the quotient greater by one than the number of means; the root is the common ratio; which being found, the means are readily obtained. EXAMPLE. Find 3 means between 37 and 2997. 2997 = 4781 = 3. {*}= Obs. In the ordinary system of Notation, the value of a figure increases ten-fold with every increase of its distance from the units' place. This is the convention in common use. But there is no reason why this should necessarily be the convention; nor why it should not be agreed that the value of a figure shall increase five-fold, six-fold, or a hundred-fold with every increase of its distance from the units' place. In these cases respectively 10 would stand for the numbers five, six, or one hundred. Def. 1. The number which 10 stands for is called the radix of the scale of notation. Def. 2. Scales are termed the binary, ternary, quaternary, quinary, senary, septenary, octenary, nonary, denary, undenary, duodenary, according as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 is the radix of the scale. Obs. As many different figures are required in any scale, as are equal to the radix of the scale. Therefore in the undenary and duodenary scales, let t stand for ten, e for eleven. I. TO CHANGE AN INTEGER FROM ONE SCALE TO ANOTHER. Rule. Divide the given number by the radix of the new scale; divide again the quotient, and repeat the division as often as possible. The successive remainders are the successive digits, the first being the unit's digit. EXAMPLES Change 814 from the denary to the senary scale; and the result to the quinary. (1) 6)814 5)3431 (2) 11. TO CHANGE A FRACTION FROM ONE SCALE TO ANOTHER. Rule. Multiply by the radix of the scale, and reduce the result to a mixed number : multiply the fractional part again by the radix, and reduce C the result: in like manner proceed as long as any fractional part remains. The several integers are the digits of the number in the new scale, the first being the digit immediately following the separating point. EXAMPLES 14 42 15 5 27 27 9 15 6 2 9 9 3 = 2 Ans. =.112 ternary. 2. Change id from the quinary to the ternary scale. 3 14 -X3= -=1 14 14 Ans. =1. 111. TO ADD, SUBTRACT, MULTIPLY, OR DIVIDE IN ANY SCALE. Rule. Perform the operations precisely as in the denary scale, carrying always the radix of the scale or its multiples, instead of 10 and its multiples. EXAMPLES. 1. Add together in the nonary scale 4567, 803, 111: and subtract in the senary scale 35432 from 43215. (1) 4567 (2) 43215 803 35432 111 3343 Ans. 5582 Ans. 2. Multiply 476te8 by 56 in the duodenary; and divide the product by 56. (1) 476te8 (2) 56)21580420(476te8 Ans. 1t0 56 2395500 lelt6t4 358 326 21580420 Ans. 820 504 542 506 380 380 |