5. The sum of the first 9 terms of an arithmetical series is 126, and the 10th term is 29. Find the sum of the first 15 terms of the series. 6. Assuming the formula for the sum of n terms of a geometrical series, whose first term and common ratio are given: if Sn represent this sum, find the value of S1+S2+S3+ . . ..+Sn. 7. The difference between the 4th and 5th terms of a geometrical series is twice the difference between the 5th and 6th terms, and the sum of the series ad infinitum is equal to 8: find the first term and common ratio. 8. The coefficient of the 5th term of (1+x)" is 6 times that of the preceding term: find the coefficient of the last term but two. 9. Show that the coefficient of the (n+1)th term of (1+x)2n is twice the coefficient of the (n+1)th term of (1+x)2n−1. 10. Expand (ax-x2) to 5 terms, and write down the general term of (a-x−3)−4. 11. Find the number of combinations of 12 things taken r together; r being such that the number of combinations formed may be the greatest possible. 12. How many different permutations can be made out of the letters in the word "Examination" taken all together; and in how many of such permutations will e be the first letter? 13. Out of 20 white and 8 black balls, how many collections can be made, each composed of 3 white and 2 black balls? 1 14. Given log 2=3010300, and log 3=4771213. Find the logarithms of 675, 00216, and (1+1)" be expanded by the binomial theorem, and ʼn be 15. If (1+ increased without limit (so that quantities of the form be neglected), to what well known function in the theory of logarithms will such expansion become equal? 16. From the series loge (1+x)=x— x2 X3 .... deduce a converging series, from which the logarithm of either of two consecutive numbers can be found when we know that of the other. 17. Define what is meant by the modulus of a system of logarithms. 18. What is the probability that if a shilling be tossed four times it will fall head once at least? 19. There are n different urns containing black and white balls; the whole number of balls in each urn being respectively m1, M2, M3, mn, and the number of white balls respectively P1, P2, P3, A ball is taken at random from one of the urns; what is the probability of its being a white ball? .... Pn. 20. From a collection of 36 balls, comprising 10 green, 12 white, and 14 red balls, 3 are to be drawn. What is the probability that two out of the three drawn will be green and the other white? 21. Find Ana when x is variable, the increment of x being unity. 22. Given log 235=2·3710679. 236 2.3729120. 23. State the respective merits of book-keeping by Single and Double Entry. 24. What are the principal books required in book-keeping by Single Entry and Double Entry respectively? 25. What do the Trial Balance Sheet and the Final Balance Sheet contain, and what is the difference between them? SECOND YEAR'S EXAMINATION, 1873. I. 1. Two lives x and y being proposed, find the probability that x will die before y 2. Two offices have each £1,000,000 assured. Office A by 1000 policies of £1000 each, and office B by 2000 policies of £500 each. Assuming all the ages equal and the rate of mortality to be 1 per cent per annum, find an expression for the probability in each case that the claims will amount to £20,000 in the course of a year. 3. Deduce a formula for finding the value of a deferred annuity certain. 4. What is meant by "force of interest"? Show how th "force" is obtained from the "rate" of interest. 5. The Guardians of a Poor Law Union are desirous of borrowing £5000; the loan to be discharged and the interest paid by an equal halfyearly charge upon the rates extending over thirty years. Assuming that the rate of interest at which the loan is obtained is 5 per cent per annum, find what the half-yearly charge should be. Given 60 at 2 per cent to be 2272836. Draw up a schedule showing what portions of each of the first four payments are applicable for the payment of interest and the discharge of the capital account. 6. Describe the ordinary form of a mortality table; and mention the tables which are generally constructed from it in order to make it available for monetary calculations. 7. Given a table of the values of annuities on two joint lives for every combination of quinquennial ages, explain how to approximate to the value of an annuity for two intermediate ages. 9. State the approximate increase to be made to the value of a life annuity payable yearly when the same is to be paid half-yearly, quarterly or monthly. 10. Given a table of annual premiums for assurances for the whole of life, how would you construct a table of half-yearly premiums? 11. Find a formula for the value of a reversionary annuity so as to return one rate of interest while the annuity is in reversion, and another rate when it is in possession. 12. What addition should be made to the value of a curtate annuity payable yearly (a), to obtain the value of a complete annuity payable half-yearly (a)? 13. A sum of money (s) is to be applied in the purchase of an annuity on three lives, x, y, z, such that the annual payment while all three are alive shall be A, when one has died A, and when only one survives A. Find the value of A. II. 14. Explain the principles upon which Mr. Woolhouse has constructed his "Improved Theory of Annuities and Assurances", and state briefly the advantage of the proposed system. 15. Investigate the formula for finding the annual premium for a contingent assurance. If asked to quote a premium, how would you proceed? 16. How would you proceed to form D and N columns; and what checks would you use to secure accuracy? 17. Show how to construct a table of premiums for the assurance of £1 for the term of one year, by a continuous process. 18. Show that Dx-Mx-(1-v)Nx-1. = 19. Required the value of a policy for £100, effected at age 40, which has been in force for 5 years and 7 months. 20. Give a form of endorsement to be placed upon a policy in order to substitute a half-yearly for an annual premium. 21. Draw up a form of Policy Register, omitting the consideration of bonus. 22. What does the following formula represent 23. A at the age of x effected an ordinary whole life assurance at a premium P; n years having elapsed he wishes to convert it into a contingent assurance, the counter life being aged y; what premium should he pay on the new policy? 24. Investigate an expression for the nth presentation to a living. 25. Find a formula to express the amount to which a life annuitydue will on the average accumulate at the end of the year of death, supposing each payment to be invested and accumulated at compound interest. JOURNAL OF THE INSTITUTE OF ACTUARIES AND ASSURANCE MAGAZINE. On the Rate of Mortality among Adult Government Emigrants on the Voyage to Australia, during the Years 1847-1861 inclusive, as determined from the Reports of the Emigration Commissioners. By J. J. McLAUCHLAN, A.F.A., of the Scottish Equitable Life Assurance Society. THE object of the following paper is to describe an investigation recently made, to determine, from observed facts, the risk incurred by persons making voyages to the Australian Colonies. This investigation was carried out under the superintendence of Mr. Sprague, at whose suggestion the following account of it was prepared. For more than thirty years past, an Annual Report has been presented to Parliament by the Emigration Commissioners; and a set of those Reports, extending from the 1st to the 25th inclusive (with the exception of the 22nd), supplied the required material. Among the other duties of the Commissioners, is that of chartering ships for the conveyance of emigrants, either wholly or partly at the public expense, to certain of the colonies; and in the Appendixes to their 8th and subsequent Reports, they have given returns of the number of ships and emigrants despatched by them annually to each of the colonies. Among VOL. XVIII. 2 D |