support the garrison; and the number of men alive when the provisions were exhausted. 22. Ans. 6 days, and 26 men remained alive when the provisions were exhausted. A Ship with a crew of 175 men set sail with a store of water sufficient to last to the end of the voyage. But in 30 days the scurvy made its appearance, and carried off three men every day, and at the same time a storm arose, which protracted the voyage three weeks. They were however just enabled to arrive in port, without any diminution in each man's daily allowance of water. Required the time of the passage, and the number of men alive when the vessel reached harbour. 23. Ans. The voyage lasted 79 days, and the number of men alive was 28. Three persons, A, B, and C, went into a gaming house; the sums which they severally had, were in a decreasing geometrical progression. Upon quitting it they found that the sums which they then had, were in a decreasing arithmetical progression; that what B had remaining was to what he had lost in proportion of the sum to the difference of what he and C had at first; and that Chad neither won nor lost. If C had won what A lost, he would then have had £64 more than A had remaining; also the whole sum which they had remaining was to that they had lost as 6:7. Required the sums which they had at first. Ans. 144, 48 and 16 pounds respectively. 24. The Fly starts 10 miles before the Telegraph; but the Fly coachman having made an appointment with the driver of the Telegraph, walks his horses so as to be overtaken at the end of the second mile. Now it is observed, that the number of revolutions made in a given time by the hinder wheel of the Fly, its fore wheel, and the hinder wheel of the Telegraph increase in arithmetical progression, and that the circumference of these wheels, viz. of the fore wheel of the Fly, its hinder wheel, and the hinder wheel of the Telegraph, increase in a geometrical progression, whose common ratio is the same as the common difference of the arithmetical progression. It is required to find the ratio that the wheels bear to each other. Ans. 1, 2, 4 are their proportional lengths. 25. A company of Merchants fitted out a privateer, each subscribing £100. The captain subscribed nothing, but was entitled to a £100 share at the end of every certain number of months. In the course of 25 months he captured three prizes, which were in geometrical progression, the middle term being one-fourth of the cost of the equipment, the common ratio the number of months which entitled the captain to his £100 share, and their sum £1375 more than the cost of the equipment. After deducting £875 for prize-money to the crew, the captain's share of the remainder amounted to one-fifth of that of the company. Required the number of merchants, and the captain's pay. Ans. The number of merchants was 25, and the captain was entitled to a £100 share at the end of every 5 months. 26. On the institution of Savings Banks, an industrious labourer with his wife and children saved each a certain number of pence in a decreasing arithmetical progression. The sum saved monthly, was less by 38. 3d. than would have purchased one-sixth of as many bushels of wheat as the seventh child saved pence: the price of wheat being such that the sum saved by the eldest and fifth child augmented by 10s. would buy two bushels. But wheat rising 2s. per bushel, and work being scarce, the family find the sum saved would not buy as much wheat as their former savings by two bushels; when it appears that at this rate the sum annually 398 Problems in Arithmetical and Geometrical Progressions. saved would be less by five guineas than by the former. Now the two youngest dying, it is found that if the remaining members of the family saved each one shilling less than the oldest child had done before the rise of wheat, their monthly account with the bank would not be affected by the deaths of the two youngest: but if they saved only 2d. less than the oldest had done, their monthly account would be 28. 1d. less than it was at the first institution. Of how many did the family consist? What were the sums saved by each? and what was the price of wheat? Ans. The family at first consisted of 10. The labourer saved 4s., each member saving 3d. less than the preceding. And the price of wheat was ss. per bushel. APPENDIX I. I. Problems in Arithmetic Progression. 1. DETERMINE the 28th term of the series 13, 124, 12, &c. 2. Having given the first and last terms of an arithmetic progression, and their common difference; determine their number of terms. 3. Having given the first and last terms of an arithmetic progression, and the number of terms; determine the progression. 4. In an arithmetic progression it is observed that the fifth and ninth terms are 13 and 25: what is the 7th term? 5. If three quantities are in an increasing arithmetic progression; show that the second will have to the first a greater ratio than the third to the second. 6. Find the sums of the following series: 1+5+9+13 + &c. to n terms. 1+4+7+ 10 + &c. to 12 terms. 5+7+9+ 11 + &c. to 50 terms. 2+2+2+3+ &c. to 13 terms. 1 +++ &c. to 16 terms. na - b + (n - 1). a + (n-2). a + b + &c. to n terms. (a + x)2 + (a2 - x2) + (a - x)2 + &c. to n terms. 8. Having given the first and last terms, and the sum of an arithmetic series; determine the common difference. 9. Having given the first term = 1, the number of terms = n, and the sum = s, of an arithmetic series; determine the common difference. |