§ 192. If we take any arithmetical series, as 3 5 7 9 11 13 15 17 19, &c. 19 17 15 13 11 9 7 5 3 by reversing the order 22 22 22 22 22 22 22 22 22 | of the terms. Here we see that the sum of the terms of these two series is equal to 22, the sum of the extremes, multiplied by the number of terms; and consequently, the sum of either series is equal to the sum of the two extremes multiplied by half the number of terms; hence, we have CASE III. To find the sum of all the terms of an arithmetical progression. RULE. Add all the extremes together and multiply their sum by half the number of terms, the product will be the sum of the series. Q. How do you find the sum of an arithmetical series? EXAMPLES. 1. The extremes are 2 and 100, and the number of terms 22: what is the sum of the series? We first add together the two extremes and then multiply by half the number of terms. OPERATION. 2 1st term 100 last term 102 sum of extremes 11 half the number of terms 1122 sum of series. Ans. 1122. 2. How many strokes does the hammer of a clock strike in 12 hours? Ans: 78 3. The first. term of a series is 2, the common difference 4, and the number of terms 9, what is the last term and sum of the series? Ans. last term 34, sum 162. 4. If 100 eggs are placed in a right line, exactly one yard from each other, and the first one yard from a basket: what distance will a man travel who gathers them up singly, and places them in the basket? Ans. 5 miles, 1300 yards. GEOMETRICAL PROGRESSION. § 193. If we take any number, as 3, and multiply it continually by any other number, as 2, we form a series of numbers, thus, 3 6 12 24 48 96 192, &c, in which each number is formed by multiplying the number before it, by 2. This series may also be formed by dividing continually the largest number 192 by 2. Thus, 192 96 48 24 12 6 3. A series formed in either way is called a Geometrical Series, or a Geometrical Progression, and the number by which we continually multiply or divide, is called the common ratio. When the series is formed by multiplying continually by the common ratio, it is called an ascending series; and when it is formed by dividing continually by the common ratio, it is called a descending series. Thus, 3 6 12 24 48 96 192 96 48 24 12 6 192 2 is an ascending series. is a descending series. The several numbers are called terms of the progression. The first and last terms are called the extremes, and the intermediate terms are called the means. Q. How do you form a Geometrical Progression? What is the common ratio? What is an ascending series? What is a descending series? What are the several numbers called? What are the first and last terms called? What are the intermediate terms called? § 194. In every Geometrical, as well as in every Arithmetical Progression, there are five things which are considered, any three of which being given or known, the remaining two can be determined. They are, 1st the first term, 2nd the last term, 5th the sum of all the terms. By considering the manner in which the ascending progression is formed, we see that the second term is obtained by multiplying the first term by the common ratio; the 3rd term by multiplying this product by the common ratio, and so on, the number of multiplications being one less than the number of terms. Thus, 3=1 1st term, 3x2 6 2nd term, 3×2×2=12 3rd term, 3×2×2×2=24 4th term, &c. for the other terms. But 2×2=22, 2×2×2=23, and 2×2×2×2=2a. Therefore, any term of the progression is equal to the first term multiplied by the ratio raised to a power 1 less • than the number of the term. Q. In every Geometrical Progression, how many things are considered? What are they? CASE 1. Having given the first term, the common ratio, and the number of terms, to find the last term. RULE. Raise the ratio to a power whose exponent is one less than the number of terms, and then multiply the power by the first term, the product will be the last term. EXAMPLES. 1. The first term is 3 and the ratio 2; what is the 6th term? 2×2×2×2×2=25=32 3 1st term Ans. 96 2. A man purchased 12 pears: he was to pay 1 farthing for the first, 2 farthings for the 2nd, 4 for the 3rd, and so on doubling each time: what did he pay for the last? Ans. £2 2s. 8d. 3. A gentleman dying left nine sons, and bequeathed his estate in the following manner: to his executors £50; his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger what was the eldest son's portion? : Ans. £25600. 4. A man bought 12 yards of cloth, giving 3 cents for the 1st yard, 6 for the 2nd, 12 for the 3rd, &c.: what did he pay for the last yard? Ans. $61,44. CASE II. § 195. Having given the ratio and the two extremes to find the sum of the series. RULE. Subtract the less extreme from the greater, divide the remainder by 1 less than the ratio, and to the quotient add the greater extreme: the sum will be the sum of the series. Q. How do you find the sum of the series? EXAMPLES. 1. The first term is 3, the ratio 2, and last term 192: what is the sum of the series? 192-3-189 difference of the extremes, 2-1=1)189(189; then 189+192=381 Ans. 2. A gentleman married his daughter on New Year's day, and gave her husband 1s. towards her portion, and was to double it on the first day of every month during the year: what was her portion? Ans. £204 15s. 3. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2nd, 9 for the 3rd, and so on to the last: what did he pay for the last bushel and for the 10 bushels? Ans. last bushel $196,83, total cost $295,24. 4. A man has 6 children; to the 1st he gives $150, to the 2nd $300, to the 3rd $600, and so on, to each twice as much as the last: how much did the eldest receive and what was the amount received by them all? Ans. Eldest $4800, total $9450. APPENDIX. MENSURATION. 196. A triangle is a figure bounded by three straight lines. Thus, BAC, is a triangle. The three lines BA, AC, BC, are called sides: and the three corners, B, A, and C, are called angles. The side BC is called the base. B When a line like AD is drawn making the angle ADB equal to the angle ADC, then AD is said to be perpendicular to BC, and AD is called the altitude of the triangle. Each triangle BAD or DAC is called a right angled triangle. The side BA or the side AC, opposite the right angle, is called the hypothenuse.* The area or content of a triangle is equal to half the product of its base by its altitude. EXAMPLES. 1. The base of a triangle is 40 yards and the perpendicular 20 yards: what is the area? We first multiply the base by the altitude and the product is square yards, which we divide by 2 for the area. OPERATION. 40 20 2)800 Ans. 400 square yards. 2. In a triangular field the base is 40 chains and the perpendicular 15 chains: how much does it contain? (see § 64.) Ans. 30 acres. 3. There is a triangular field of which the base is 35 rods and the perpendicular 26 rods: what is its content? Ans. 2A. 3R. 15P. |