and the value of D is similarly found to be D and hence by [4] art. (562) The given series is the value of ax (496) when k = 1, or of exy, where x is the Napierian log. of y; and hence the series found for x is just the series for the Napierian log. of y or ly in terms of y, and coincides with the series for L' (1+) in art. (514) when y is taken in that series for 1+x, and consequently y-1 for x, and instead of L'; for then L'e le=1, and L' (1+x)=ly. = APPLICATION OF LOGARITHMS TO COMPOUND INTEREST AND CERTAIN ANNUITIES. I. COMPOUND INTEREST. (564.) As in compound interest, interest is chargeable not only on the principal, but also on the interest as it falls due, the amount therefore becomes a principal at the beginning of each period of payment.* t = time or number of payments, r = ... amount of £1 for one of the periods of payment, then is r = £1+ the interest of £1 for one of the periods of payment. (565.) The interest of £1 for one period will be the rate per cent. divided by 100. Thus, if the rate per cent. be 5, and the period be Ì year, the interest of £1 for 1 period is 05; and therefore r 1·05. So for a rate of 4 4 per cent. 1+ =1+·04 = 1.04. 100 When the interest is payable half yearly, that is, when one period is half a year, and the rate is 5 per cent. per annum, or 2.5 2.5 for one period, r = 1 + =1+·025 = 1.025. If the 100 rate is 4 per cent., and the period a quarter of a year, 1 r = 1+100 rate or period. =1+011.01; and so on for any other Since r is the amount of £1 for one period, and since any two sums are proportional to their amounts for the same rate and time, if a' be the amount of p pounds for one *No definitions are given here of the terms-principal, amount, rate per cent., and present value, as these are to be found in common treatises of arithmetic. period, then the following proportion is evidently true, namely, hence a pr, 1:p=r:a'; that is, (566.) The amount of any principal for one period is equal to its product by the amount of £1 for the same time." Therefore pr.r=pr2 = amount of pr pounds for one period, or p pounds for two p p ... ... (567.) COR.-When p= 1, art; and hence for t=I, 2, 3, 4, a is = r, r2, p3, p4, ... of £1 for one period; 2 is its ... that is, r being the amount amount for two periods; p3 for three periods, and so on; and generally rt is its amount of t periods. In [6] the sign of ratio (:) is used for that of division. (568.) It appears from these formulas, that when any three of the four quantities p, a, r, and t, are given, the fourth can be found. (569.) Since the amount of a sum of money p for the time t is a prt, therefore p is also the present value of a sum a, which is to be paid at a future time distant from the present by t intervals, and by [2], p = α EXAMPLES. 1. Find the amount of £600 for 10 compound interest. La Lp +tLr L 600 + 10 L 1-042-7781513+ 10 x 01703332-7781513+17033302.9484843 = L 888.146. Hence a = £888·146 — £888, 2s. 111d. 2. In what time would a sum be doubled at 5 per cent. compound interest? 5 Here r=1+ = 1.05, and p may be assumed = 1, 100 then a 2; and hence = t={La-Lp}: Lr={L2—L1}: L1-05 ⚫3010300 ⚫0211893 = 14.207 years 14 years 75 days. EXERCISES. 1. Find the amount of £500 for 21 years, reckoning compound interest at 4 per cent. Ans. £1260, 2s. 5d. 2. What is the present value of £400, due 3 years hence, at 5 per cent. compound interest? Ans. £345, 10s. 83d. 3. Find the amount of £225 for 25 years at 4 per annum compound interest, the half yearly. per cent. interest being payable Ans. £605, 12s. 1 d. 4. In what time will £600 amount to £5400, the interest being compound at 5 per cent.? Ans. 45-033 years. II. CERTAIN ANNUITIES. (570.) 'An annuity is a constant sum of money paid at regular intervals; and it is said to be certain, when it is independent of accidental circumstances.1 1. AMOUNT OF AN ANNUITY. (571.) When the annuity is in arrears, that is, when the payment has been forborn beyond the period when it is due." t = the time or the number of payments due, r = the amount of £1 for one period. Then since the last annuity is due only at the expiry of the time t, no interest is chargeable upon it, so that its amount is only a; the annuity preceding it has been due for one period, and hence (566) its amount is ar; the annuity preceding this has been due for two periods, hence its amount is ar2; the amount of the next preceding is evidently ar3; of the next art; and so on to the first, which has been due for (t—1) periods, and its amount is therefore art1. Hence the amount of all the annuities is = a + ar+ar2 + ar3 + But this is just an equirational series, ... ·art-1 and hence (431) its Im La+L(rt — 1) — L (r—1) The last formula is found thus: 1 1 r 1 [1] ... [2] |