[N. B. The result of every equation is not a whole number.] 5. A travels at the rate of seven miles in five hours, B sets off from the same place eight hours after, and travels the same road at the rate of five miles in three hours; how long and how far will A travel before he be overtaken by B? 6. A person lays out a certain sum of money in goods, which he sold again for £24, and gained as much per cent. as the goods cost him. What was the sum laid out? 7. Find the sum of seven terms of the series 3 + 10 40 2 &c. ad infi 10 102 103 104 105 nitum. Insert two arithmetic means between 2 and 7; and prove an arithmetic mean greater than a geometric. 8. How many permutations can there be in the letters of the word ALGEBRA? whole number, will always be a whole number, whatever be the number of terms. 10. Write down three terms of a-x, and the cube of x2+y+z. 11. Divide a2— x2 by a+x, and bring the fraction to its lowest terms. 12. Investigate the rule for extracting the square root of a binomial, one of whose factors is a quadratic surd, and the other rational; and apply it to find the root of 7-13. 13. Prove the rule for discovering when the multiplication of two numbers is correct, by dividing the multiplicand," multiplier, and product, by 9. 14. Express 359, when the local value is 6; and prove the rule. 15. Find the greatest and least positive integral values of x and y in the equation 3x+5y=68. 16. What is the present worth of £130, due fourteen months hence, allowing 4 per cent. simple interest? 17. A person purchased an annuity of £160 per annum for ten years, to commence at the end of seven, and payable half yearly. After three of the seven years have expired, what is its value allowing 5 per cent, compound interest? Express the result loga rithmically. an integer or a fraction, according as (n) is odd or even. a+b 4. Insert (m) harmonic means between (a) and (b). 5. If, between all the terms of an arithmetical progression, the same number of arithmetic means be inserted, the new series will still form an arithmetical progression. 6. Sum the following series: 2 + 5+ 8 + &c. to (s) terms. + + + &c. ad infin. 1 + 5 + 13 + 29 + 61 + &c. to (n) terms. 7. What is the number of permutations of (n) things, of which there are (p) of one sort, (q) of another, and () of a third. And shew that the total number of combinations which (n) things admit of, = 2" 1. 8. A, who travels only every other day, sets off from a certain place nine days after B, in order to overtake him, but travels four times as fast as B does. When will they come together? 9. Divide the number 35 into two such parts, that the square of the less divided by the difference of the two parts = 45. 10. A person owes £150, to be paid at the end of nine months; and £60, at the end of six months. Required the equated time of payment, and investigate the rule. 11. a b is a ratio where (a) is prime to (b). They are the least in that proportion-and shew that a +x a very nearly in the proportion of a +mx: a, when (x) is small compared with (a). 12. Required the square root of 5-4/3, and of 9-4-3. 13. Required the discount upon £100, due three months hence, at 4 per cent. per annum. 14. Find the present worth of an annuity (A) to continue for (») years at simple interest. 15. What is the present worth of an annuity of £60, to commence in two years, and to continue for ever at 3 per cent. per annum. 16. Given the log. of 8.1213.9096256. It is required to state the log. of $12.13; and of .81213; and of .081213. 17. Resolve 17 into a continued fraction. 18. Required a number such, that, divided by 5, 4, 3, respectively, it may leave remainders 2, 3, 4, respectively. 19. The difference between any number, and that number inverted, is divisible by the local value minus one. 1817. 1. Find the sum of the coefficients of a binomial raised to the nth power; write down the pth term of the binomial, and deduce from thence the 4th term of (ab). 2. Prove that when four quantities are proportional, the product of the extremes = the product of the means; and convert === + -y 1 59] 3y: 11; 5. 8. xy + x + y = 19. x2.x+4+2x.x+4=2 x+4 x + 2)2 + 2√x.x + 2 − 3√ √ x = 46 + 2x. 4. Find a sum consisting of P pounds, Q shillings, the double of which shall be Q pounds, P shillings. 5. Find two numbers in the proportion of 9 to 7, so that the square of their sum shall equal the cube of their difference. 6. A person being asked the hour of the day, answered thus: if of the number of hours remaining till midnight be multiplied by 4, the product will as much exceed 12 hours, as of the present hour from noon is below 4. What was the hour after noon? Ꮖ 7. What number is that, which being divided into any two parts, and y, x2 + y = y2 + x ? s. The sum of 7 numbers in arithmetical progression = 28, and the sum of their cubes = 784. Determine the progression. 9. Find 4 numbers in arithmetical progression, which being increased by 2, 4, 8, and 15 respectively, the sums shall be in geometrical progression. 10. Which is the greater, a geometrical or an arithmetical mean; and by what quantity does the greater mean exceed the less? 11. Prove that the reciprocals of quantities in harmonical progression are in arithmetical progression, and find a fourth harmonical proportional to 6, 8, and 12. 12. Prove that of n things, r of them being always taken together, the number of permutations= n. (n-1). (n-2).... (n-r+1); and that the number of combinations = and find the number of permutations which can be made of the letters in the word "examination," two of them being always taken together. 13. Sum the following series: (1.) 8 + 15 + 22+ &c. to 12 terms. (2.) 116+ 108 + 100+ &c. to 10 terms. + &c. ad inf. 1 + (5.) 3 + 7 + 15 + &c, to n terms. 14. If ab be prime to c, a, and b are each of them prime to c. 15. If the sum of the odd digits of any number N, whose local value is r, be equal to the sum of the even digits, teger. Required a proof. N r+1 is an in 16. Find in what time £600 will double itself at simple interest; and also find the difference in the interest of £1200 put out at 5 per cent. for ten years, at simple and at compound interest. 17. The discount of £500 due four years hence £500 :: 16; required the rate of interest. 18. £600 is due eight months hence, £500 nine months hence, and £1200 at present. Find the equated time of payment by the common method, and shew wherein the error of the process consists. 19. A person borrowed £4000. In what time will he be out of debt, supposing him to vest £200 at the end of every year, till the whole be paid off; the rate of interest being 5 per cent. per annum, simple interest? 20. Extract the square root of 2 √ 2-1. 3 2,743, and 21. In how many ways can £40 be paid with half-guineas and pistoles; the value of the pistole being 17s.? ALGEBRA. PART II. 1820. 1. IF (a) be a root of the equation x" + pTM− + qx”¬3 ÷. +r+u=0; prove that (-a) is a divisor of the equation without assuming its resolution into factors. 2. Transform the equation a3 which shall want the third term. 4x+5x 20 into one 3. If (P) be the greatest negative coefficient of any equation (m) the number of terms preceding the first term whose coefficient is negative, prove that 1+ P is a superior limit of the roots of the proposed equation. 4. The roots of the equation na" + (n − 1) px”→2 + are limits between the roots of the equation " + paTM1.+ when the roots of the latter equation are possible. =0 ..=0 |