Ex. 1. The first term in a geometrical series is 2, the last term 250, and the number of terms 4; what is the ratio? 250 2: = 125; 4—1=3; and 3/1255, Ans. 2. The Extremes are 3 and 48, and the number of terms 3; what is the ratio? Ans. 4 or 4. 3. The extremes are 3 and 243, and the number of terms 5; what is the ratio? 377. PROBLEM 3. To find the sum of a series, the extremes and ratio being given. Having a series given, e. g. 2, 10, 50, 250, 1250, 6250, mul tiply each term except the last by the ratio, 5; thus, 2, 10, 50, 250, 1250, 10, [6250], 6250; Given series, Product by 5, and we shall evidently form a new series like the old, except the first term of the old is not found in the new. Now, if the old except the last term be subtracted from the new, the remainder will be the difference of the extremes in the old series the other terms in the two series canceling each other; the remainder will alsc be 4 times the sum of all the terms except the last in the old series; for once a series from 5 times a series must leave 4 times the series; .. of this remainder plus the last term must be the sum of all the terms in the old series; but 4 is the ratio less 1. A similar explanation is always applicable. Hence, RULE. Divide the difference of the extremes by the ratio less one, and to the quotient add the greater extreme. Ex. 1. The extremes are 2 and 486, and the ratio 3; what is the sum of the series? 4862484; 3 — 1 = 2; 484 ÷ 2 = 242; and 2 12 + 486728, Ans. 2. The extremes are 4 and 5184, and the ratio 6; what is the sum of the series? 3. What debt will be discharged by 12 monthly payments, the 1st payment being $1, the 2d $2, and so on in a geometrical series? 877. Object of Frocem & ANNUITIES. 378. AN ANNUITY is a sum of money payable annually, or at any regular period, either for a limited time or forever. An annuity is in arrears when the installments remain unpaid after they are due. The AMOUNT of an annuity in arrears is the interest of the unpaid installments added to their sum. 379. PROBLEM 1. To find the amount of an annuity in arrears, at simple interest. Ex. 1. An annuity of $100 per annum has remained unpaid 4 years; what is its amount? Ans. $436. The 4th payment is due to-day and is worth just $100; the 3d payment due 1 year ago is worth $106; the 2d payment due 2 years ago is worth $112; and the 1st payment due 3 years ago is worth $118. But these numbers, $100, $106, $112, and $118, are in arithmetical progression. Hence, RULE. Find the last term of the series by Art. 369, and the sum of the series by Art. 372. 2. Purchased a farm for $5000, agreeing to pay for it in 5 equal annual installments; the 5 years having elapsed without any payment being made, what is now due, allowing simple interest? Ans. $5600. 3. A salary of $600 per annum is in arrears for 8 years; to what does it amount, allowing simple interest at 7 per cent.? 380. PROBLEM 2. To find the amount of an annuity in arrears at compound interest: Ex. 1. What is the amount of $1 annuity, per annum, in arrears for 3 years, at 6 per cent. compound interest? The 3d instalment becoming due to-day, is worth just $1; the 2d having been due 1 year, is worth $1.06; and the 1st having 378. What is an Annuity? When is an annuity in arrears? What is the Amount of an annuity? 379. Object of Problem 1? Rule? 380. Problem 2? been due 2 years, is worth $1.1236; .. $1 - $1.06 + $1.1236 = $3.1836, the sum sought. But these numbers are in geomet rical progression. Hence, RULE 1. Find the last term of the series by Art. 375, and the sum of the series by Art. 377; or, RULE 2. Multiply the amount of $1, found in the following table, by the annuity, and the product will be the required amount. TABLE, Showing the amount of the annuity of $1, £1, etc., at 4, 5, 6, and 7 per cent. compound interest, from 1 to 20 years. 2. What is the amount of an annual salary of $1000, in arrears for 5 years, at 6 per cent.? Ans. $5637.093. 3. What is the amount of an annual rent of 100£, in arrears for 15 years, at 5 per cent.? Ans. 2157.8564£= 2157£ 17s. 1d. 2qr. 4. What is the amount of an annual pension of $500, in arrears for 12 years, at 6 per cent.? 380. 1st Rule? 2d Rule? PERMUTATIONS. 381. PERMUTATION is the arranging of a given number of things in every possible order of succession. 382. PROBLEM. To find the number of permutations of a given number of things. The single letter, a, can have but 1 position, i. e. it cannot stand either before or after itself; the 2 letters, a and b, furnish the 2 permutations, = ab, the number of which is expressed by the product of bas' α × 2 = 2; and if a 3d letter, c, be introduced, we have cab, c ba a cb, b c a ; i. e. the new letter, c, may stand 1st, 2d, or 3d abc, bac) in each of the 2 permutations of a and b; hence the number of permutations of 3 things is expressed by the product, 1 × 2 × 3 6. If a 4th letter,d, be taken, it may stand as 1st, 2d, 3d, or 4th, in each of the 6 permutations of a, b, and c, and, of course, furnish 4 times 6 = 1 X 2 X 3 X 4: 24 permutations. = By the above, it is evident that the number of permutations RULE. Form the series of numbers, 1, 2, 3, 4, etc., up to the number of things to be permuted, and their continued product will be the number of permutations. Ex. 1. How many different integral numbers may be expressed by writing the 9 significant digits in succession, each figure to be taken once, and but once, in each number? Ans. 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 = 362880. 2. In how many different orders may a family of 10 persons seat themselves around the tea table? 381. What is Permutation? 382. Object of the Froblem? Rule? MENSURATION. 383. MENSURATION is the art of measuring lines, surfaces, and solids. The principles are all Geometrical, and are very numerous. A few only of the more simple are here presented. 384. Two parallel lines are everywhere equally distant from each other. When two lines meet so as to form equal angles, the lines are perpendicular to each other and the angles are right angles. A right angle contains 90°. An acute angle is an angle of less than 90°. An obtuse angle is an angle of more than 90°. Two lines are oblique to each other when they meet so as to form acute or obtuse angles, and the angles are oblique angles. 385. A TRIANGLE is a plane figure which is bounded by three lines. The base of a triangle (or any other figure) is the side on which it is supposed to stand. The altitude of a triangle is the perpendicular distance from the angle opposite the base to the base, or to the base extended. 386. PROBLEM 1. To find the area of a triangle. RULE. Multiply the base by half the altitude. Ex. 1. The base of a triangle is 7 inches and the altitude 8 inches; what is its area? Ans. 28sq. in. 2. The base is 8ft. and the hight 11ft.; what is the area? 383. What is Mensuration? 384. What of two parallel lines? What is a right angle? An acute angle? Obtuse angle? What are oblique lines? Oblique angles? 385. What is a Triangle? Its base? Its altitude? 386. Rule for finding its area? |