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.or 1 year, as before; then that root of it which is denoted by the aliquot art, will be the amount of 17. This amount being multiplied by the principal sum, will produce the amount of the given sum as required.
2d. When the time is not an aliquot part of a year. Reduce the time into days, and take the 365th root of the amount of 17 for 1 year, which will give the amount of the same for 1 day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount, being multiplied by the principal sum, will produce the amount of that sum, as in the former cases.
ANNUITY is a term used for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual payments.
Annuities are divided into those that are in possession, and those in reversion : the former meaning such as have already commenced; and the latter such as will not begin till some particular event has happened, or till after some certain time has elapsed.
When an annuity is forborne for some years, or the payments not made for that time, the annuity is said to be in arrears, or in reversion.
An annuity may also be for a certain number of years; or it may be without any limit, and then it is called a perpetuity.
The amount of an annuity, forborne for any number of years, is the sum arising from the addition of all the annuities for that number of years, together with the interest due upon each after it becomes due.
The present worth, or value, of an annuity, is the price or sum which ought to be given for it, supposing it to be bought off, or paid all at once.
Let a = the annuity, pension, or yearly rent;
n = the number of years forborne, or lent for;
the amount of 1l for 1 year;
sent values of a due at the end of 1, 2, 3, n, years respectively. Therefore, the sum of all these will be the present value of the n years' annuities; and if n be infinite, it will be the present value of a perpetual annuity of a £ per term.
Now by summing this geometrical series, we have v = + + +...+ R R2 R3
the present value of the annuity which is to terminate in n
years; and if n be infinite, v =
for the value of the annuity in per
Again, because the amount of £1 in n years is R", the increase in that time is R— 1; but its amount in one year, or the annuity answering to that increase
is R 1 and as these are in the ratio of a to m, we have m =
the several cases relating to annuities in reversion are easily found to be as
In theorem (3), r denotes the present value of an annuity in reversion, after p years, or not commencing till after the first p years; and it is found by taking
n years and p years. The other formulæ are derived from those in compound interest taken in connexion with the fundamental theorem deduced above.
However, for practical purposes the amount and present value of any annuity for any number of years, up to 21, will be most readily found by the two following tables. In works professedly devoted to the subject, these tables are carried to a much greater extent.
The Amount of an Annuity of 17 at Compound Interest.
To find the amount of any annuity forborne a certain number of years.
Take the amount of 17 from the first table, for the proposed rate and time; then multiply it by the given annuity; and the product will be the amount, for the same number of years, and rate of interest. Also, the converse to find either the rate or the time.
Ex. To find how much an annuity of 501 will amount to in 20 years, at 31⁄2 per cent. compound interest.
On the line of 20 years, and in the column of 31 per cent. stands 28.2797, which is the amount of an annuity of 17 for the 20 years. Then 28.2797 × 50, gives 1413·9857 = 14137 19s Sd for the answer required.
To find the present value of any annuity for any number of years. Proceed here by the second table, in the same manner as above for the first table, and the present worth required will be found.
Ex. 1. To find the present value of an annuity of 50l, which is to continue 20 years, at 3 per cent. By the table, the present value of 17 for the given rate and time, is 14-2124; therefore 14 2124 × 50 = 710·62l, or 710l 12s 4d, is the present value required.
Ex. 2. To find the present value of an annuity of 207, to commence 10 years hence, and then to continue for 11 years longer, or to terminate 21 years hence, at 4 per cent. interest. In such cases as this, we have to find the difference between the present values of two equal annuities, for the two given times; which, therefore, will be done by subtracting the tabular value of the one period from that of the other, and then multiplying by the given annuity. Thus, the tabular value for 21 years is 14 0292, and that for 10 years is 8·1109. Then, the difference 5.9183 multiplied by 20 gives 118-3667, or 1187 7s 34d, the answer.
SERIES BY SUBTRACTION.
THIS method is most readily applicable to the cases where the several terms of the series are the differences (or the same multiple of the differences) between two equi-distant corresponding terms of some other series. A few simple examples will sufficiently illustrate the practice, whilst the principle of these processes is self-evident. The only difficulty is to find the series whose differences are the terms of the given one; and for this no general and simple rule exists.
Note. The upper line (carried to any extent) contains one term at the end, under which there is no term in the lower line. But in the above examples this circumstance creates no difference, since the terms being infinitely distant, and continually converging towards 0, that last term itself is virtually 0. However, if the law of the series be such, that the terms converge towards a limit different in value from 0, then this value being that of the last term of the upper line, must be added to the sum obtained as above.
Ex. 9. Thus, if + + + were purposed to be summed; the
2.3 3.4 4.5
process would be, if performed according to the preceding type,
But as the values of the several terms of the series, whose value is s, converge towards 1, the uncompensated term itself is 1, which added to the value already found, gives 1 — for the true sum of the series. This final term, in such cases, may be called the correction of the sum.
Ex. 4. The sum of n terms of a + 2ar + 3ar2 + 4ar3 +
*This and such questions differ from the preceding in no respect but this: that the expression for the nth term must be taken as the last of the assumed series instead of the infinitely distant term towards which in that case the succeeding terms more directly point. The correction for this must be made as directed in the note above.