5 In this case the payment for every £100 at each half year is and for every pound .025; Hence the amount = 250 (1.025)20. 2' Note. It is not strictly correct to call the interest in the last question 5 per cent. per annum. For it is evident that the amount of £100 for 100 (1.25)2 1 year is The true rate per cent. per annum is therefore 5.0625 or 5%. 1 6. If £p be borrowed at r' per cent. compound interest, and £q be repaid every year, in how many years will the debt be paid off? Let n denote the number of years required; Then the present value of the annuity of £q for n years must be £p. Art. (189). This is also evident from another consideration, for when q=pr, Y is exactly the interest of p, and it must always be paid, the debt still remaining the same. Examples LXXVII. 1. Find the simple interest of £150 for 2 years at 41 per cent; also the amount of £1015 for 9 months at 3 per cent. 2. What sum placed out at simple interest at 4 per cent. per annum on the day of a child's birth will amount to £1000 on the day that he is of age? 3. Work the last question with compound interest instead of simple. 4. Find general expressions for the Principal in terms of the interest, rate, and time; Principal amount, rate, and time; amount, rate, and time; amount, interest, and rate; in the case of simple interest. 5. Find the same general expressions as in (4) in the case of compound interest. 6. Find the amount of £1200 in 7 years at 5 per cent. comp. int. 7. Find what sum will amount to £109 198. 1d. in 4 years at 5 per cent. 8. Find at what rate per cent. a sum of money will double itself in 14 years at compound interest. 9. What sum put out to compound interest for 3 years at 5 per cent. will amount to £115 15s. 3d.? 10. At what rate of compound interest must a sum be placed out so as to amount to t times itself in n years? 11. At what rate per cent. will £118 58. produce £36 18. 4d. at compound interest in 6 years? 12. Find the interest of £350 for 5 years at 2 per cent. compound interest, the payments being made four times a-year. 13. Find the amount of £1000 at interest for 21 years at 5 per cent. ; 10 at simple interest; 2° at compound interest; 3° at compound interest payable half-yearly. 14. A person lent £1200 on condition of his being paid back £1600 at the end of three years. What interest did he receive for his money, reckoning compound interest? 15. A merchant borrows £P at r per cent. per annum, and by trading makes it produce r' per cent. What sum will he have gained at the end of a years? 16. From a capital of £10,000 yielding 5 per cent. compound interest, £800 is annually taken away. What capital will be left at the end of 10 years? 17. If a sum of £a be borrowed at r per cent. and £b be repaid at the end of each year, in how many years will the debt be discharged at compound interest? Interpret the result when b = α 100 18. A person who owes a debt of £a wishes to extinguish the debt by b annual payments. Find amount of each payment, reckoning compound interest at 4 per cent. per annum. 19. The amount of a sum of money in 4 years when the payments are annual is the same as the amount of the same sum in 3 years when the payments are quarterly. Find the rate per cent. 20. If a sum of money at a given rate of compound interest amount to m times its original value in a years, and to n times its original value 21. At what rate per cent. would the present value of a debt of £1350 payable in 5 years be the same as that of a debt of £1200 payable in 3 years? 22. A debt of £1000 is to be discharged by monthly payments of £50. If it were proposed to pay the whole of this sum at once, when should the payment be made? 23. If £10,000 be given for a perpetual annuity of £300 a-year, what is the rate of interest? 24. If in the last question the annuity continue for 30 years, what is the rate of interest? 25. A gentleman wishes to divide £10,000 between his two sons; to one he gives £5,000 down, and to the other a perpetual annuity. Find how much it ought to be, interest being reckoned at 4 per cent. 26. At what rate per cent. will an annuity of 100 guineas amount to £1534 19s. 10 d. in 12 years? 27. Find the amount of an annuity of £50 for 7 years at 5 per cent. 28. Find the amount of an annuity of £150 for 10 years, payable quarterly, at 5 per cent. per annum. 29. Find the present value of an annuity of £300 for 20 years at 4 per cent. 30. Find the present value of an annuity of 100 guineas, to begin in and to continue for ever, at 4 per cent. 10 years 31. Find the difference between the present value of an annuity of La to continue for n years, and the same annuity to continue for ever, the rate per cent. being r. Find 32. The present value of an annuity of £50 for 10 years is the same as the present value of an annuity of £20 to continue for ever. the rate of interest. 33. An annuity of £a is to commence at the end of p years and con tinue for q years. Find the equivalent annuity to commence at once and continue for ever. 34. An annuity of £200 is to begin in 14 years and continue for 7 years. Find the present worth of it; also the equivalent perpetual annuity to begin at once. 35. A gentleman wishes to secure a sum of £200 a-year for the education of his son; the payments to begin when he is 14 and to continue till he is 21. What annual sum paid for 14 years from the day of his birth will secure this? 36. What should be the purchase-money for a fee-simple of £500 a-year, the purchaser to take possession at the end of 10 years, so that he may make 5 per cent. of his money? 37. A property yielding £a per annum is given by a testator, 1° to a charity for m years; 2° to another charity for n years, and after that to a third charity for ever. Find the value of each bequest. 38. Which is the better interest, 5 per cent. payable quarterly, or 5 per cent. payable yearly? 39. If a lease of an estate for 21 years be bought for £10,000, and the annual rental be £750, what rate per cent. does the purchaser receive for his money? 40. If 7 years of a lease of 21 years have lapsed, and the whole rental be £560, subject to a reserved rent of £25, what ought to be given for the renewal of the lease, interest being reckoned at 4 per cent? CHAPTER XXI. SUMMATION OF SERIES, PILES OF BALLS, SHELLS, &c. 191. In this Chapter some examples of the summation of series will be given, in which the series are of different forms from those which have been already treated of. i. To sum the series (a + b) (2a + b) (3a + b) + (2a + b) (3a + b) (4a + b) +, &c., to n terms. Let (a+b) (2a + b) (3a + b) (4a + b) +(2a + b) (3a + b) (4a + b) (5a + b) + (na + b) (n + 1 a + b) (n + 2a + b) (n + 3 a + b)) = S. + (n − 1 a + b) (na + b) (n + 1 a + b) (n + 2 a + b) = { S+ b (a + b) (2a + b) (3a + b) (na + b) (n + 1 a + b) (n + 2 a + b) (n + 3 a + b). By subtracting in order, the corresponding lines of the lower series from those of the upper, we get = { (na + b) (n + 1 a + b) (n + 2a + b) (n + 3 a + b) - b (a + b) (2a + b) (3a + b). Hence, the sum required (na+b) (n+1.a+b)(n+2.a+b)(n+3.a+b)−b(a+b)(2a+b)(3a+b) By subtracting, in order, the terms of the lower series from the corresponding terms of the upper series, we get |