3. What is the difference between the 11th and 81st terms of the scries in question 2nd ? Ans. 1208925819614629174705152. 4. What is the sum of the series &c. continued for ever? Ans. 5. If the series 1, 3, 9, &c. be carried to 13 terms, how much will the last term exceed the 3d ? Ans. 531432. 6. What is the sum of the series 9, 6, 4, &c. continued for ever ? Ans, 27. 7. What is the number of acres in an estate, which if sold on the principles of geometrical progression, a farthing being given for the first acre, and a penny for the second, the price of the last acre will be $10,000 ? Ans. 12,59731. very nearly. 8. It is agreed to purchase ships of war, of the first rate, on the principles of geometrical progression, the price of the first ship being fixed at 15s, and of the last at £617,657 : 5s. What will the second ship cost? Ans. 5 guineas. 9. Given the sum of three numbers in geometrical progression 39, and the difference between the extremes 24. Required the numbers. Ans. 3, 9, 27. 10. Supposing ten persons to be living at the end of the first age, ten times as many at the end of the second, &c.; mankind thus decupling themselves every age. Required the number of persons living at the deluge, supposed to have happened at the end of the sixteenth age. Ans. 10,000,000,000,000,000. 11. Required four mean proportionals between 5 and 160. Ans. 10, 20, 40, and 80. 12. What four numbers in geometrical progression are they, whose sum is 24, and the sum of whose squares is 164 ? Ans. 9, 7, 5, 3. 13. Given the sum of the first and second terms of four numbers in a geometrical progression equal to 12, and that of the third and fourth equal to 108. Required the numbers. Ans. 3, 9, 27, 81. 14. The sum of four terms in a geometrical series exceeds the ratio by b, and the first term is jíz. Required the remaining terms. Ans.is, , . 15. The sum of three numbers in a geometrical progression is 13, and the mean is to the difference between the extremes, as 3:8. Required the numbers. Ans. 1,3, 9. 16. Given the difference between the first and second terms of four numbers in a geometrical progression, equal to 54 ; and the dif. ference also between the third and fourth, equal to 6. Required the numbers. Ans. 81, 27, 9, 3. 17. To find four pers in geometrical progression whose sum is 15, and the sum of their squares 85 ? Ans. 1, 2, 4, and 8. 10 ON SURDS. (128.) When a magnitude or number cannot be expressed in finite terms without the help of a fractional index, it is called a SURD: thus the square root of 2, the cube root of 3, the nth root of a + b, the cube root of (a + x)', &c. &c. may be expressed either by 12, 33, va + b, y(a + x)', &c. or by 21, 31, (a+b), (a+x);, &c. Note.-.-The precise value of these quantities cannot be ascertained; it can only be expressed by means of decimals or serirs which do not terminate ; and in this sense they are called irrational, to distinguish them from all other quantities wbatever, integral or fractional, whose values are determinate, and which are therefore denominated rational. Surds in their rudicul form, when properly reduced, are subject to all the ordinary Rules of Arithmetic. The Reduction of Surd quantities. CASE I. 129.) A rational quantity may be reduced to the form of a surd, by raising it to the power denoted by the root of "he surd. Example: . 1. Reduce 3 to form of the square root, and it becomes 138 or „9. 2 3 /23 8 2. Reduce 3 39 3. Reduce a+b. ....square root,.... v(a+b). cule root, V or 27 (130.) Surd quantities of different indices are reduced to equivalent ones with the same index, by bringing their fractional indices to a common denominator. Examples. 1. Reduce at and at to surds of the same index. 1 '' a>= ct=%d', which are surds with the same ai=22,) index 0. 6 1 and as= ai= v 2. Reduce 31 and 54 to surds of the same index. 1 2 and 6 ز The fractions and , reduced to a common denominator, arc it Now 3*=*3*=V81 ; and 5*=35=7125. Ex. 3. Reduce at and i} . Ans. vus and vbs. to Surds Ex. 4. Reduce ci and di Ans. $cs and do. Ex. 5. Reduce 3 32&2 75 Ans. 3/4 & 2 V 125. & Ex. 6. Reduce 41 and 154 Ans. /256 & 73375. 15 6 6 with the same index. 6 6 CASE III. (131.) Surd quantities are reduced to their simplest form, by observing whether the quantity under the radical sign contains a power corresponding to the given surd root, and then extracting that root. Note.-The quantity without the radical sign is called the co-efficient of the surd; and it is evident, that this quantity may always be put under the radi. sal sigo, by raisinno in ce the power denoted by the index of the Thus, 792x=(by Case I.) ✓ 7a x 7a x 2x. *749a® X //2r=_98aRx. Also, * /2a-r=vr x 2a-. =/x* (2-x)=2ar-x. CASE IV. (132.) If the quantity under the radical sign be a fraction, it may be reduced to an integral form by the following Rule.--Multiply the numerator and the denominator of the fraction by such a quantity as will make the denominator a complete power, corresponding to the root; then extract the root of the fraction whose numerator and denominator are complete powers, and take it from under the radical sign. Examples. The Fundamental Rules of Arithmetic applied to Surd Quantities. (133.) The Addition and Subtraction of Surd Quantities. Rule.-Reduce the quantities to their simplest form; and if the surd part be the same in both, then their sum or difference will be found by taking the sum or difference of their co-efficients, and annexing the common surd to the result. Examples. 1. Find the sum and difference of 716a®x and 74a®z. By Case III. ✓16aRx = 4a x, and n 4aRx = 2a vX; .. the sum =4a 7x+ 2a x=(4a+2a) x vr=6ax. the difference=40wx-20/=(4a-2a) * vx=2a vit. 2. Find the sum and difference of 3/ 192 and 3/24. and 3 24=38 X 3 = 233; 3. Find the sum and difference of 8 27 and difference of Vand V andą, reduced to a common denominator, 8 The two fractions 27 48 27 are and 162 162 |