SIMPLE INTEREST.
5
:lod is discharged
king it ought not
which put out at
to the sum due;
:ent, is equivalent
the same case as
s case corresponds
with 5 the amount,
; have by Art. 6,
r equally here, if we
, for principal and
ie present value and
or the present pay-
, interest 5 per cent ?
; .05
25 s
20.988— yip i n
4.012 —£4 0 3
discount.
;ount, but in the mer-
>unt the interest of the
3, by ■which mode more
1, is ins, and therefore
ins f 7i 2 S
1 + in 1 + in
In the example given above, 4.0625 = £4 1 3 is the sum that a
banker would receive for discounting the same bill at the above rate of
interest.
ON ANNUITIES AT SIMPLE INTEREST.
13. An annuity is a periodical income arising from lands, houses,
money lent, pensions, &c.
When the possession of an annuity is not to be entered upon until
the expiration of a certain period, it is called a reversionary or deferred
annuity; when the time of possession is not deferred, the annuity is
sometimes called immediate, but in general it is simply termed an
annuity.
At the time of acquiring the title to an annuity the party is said to
enter on possession; one of the equal intervals at which the annuity is
payable, is always supposed to elapse between the time of entering on
possession and the first payment of the annuity.
14. The amount of an annuity in a given time is the sum of all the
payments with their interest from the time of becoming due, until the
expiration of the term.
Make s = the amount of the annuity,
a — the annuity,
n — the number of years,
i = the interest of <£T for one year;
then if the annuity be £l per annum forborne n years, the last or
n th payment being received at the time it falls due, there is no interest
on it, the amount therefore is ¿£l only; the last payment but one, on
which one year’s interest is due, amounts to 1 + i; the last but two, on
which two years’ interest is due, amounts to 1 + 2 i; the last but three
to 1 + 3 i; and so on till we come to the first payment, which being
payable at the end of the first year, has (n — 1) year’s interest due
thereon, and amounts to 1 -f (?i — 1) i; the following series is there
fore the amount of an annuity of £l in n years:
1 + (1 + 0 + (1 + 2 i) + (1 + 3/) + (1 + 40 +
+ {1 + (« — 3) + { 1 + (« ~ 2) + {1 + (« - 1) i.}
This series, in which the difference between each term and the next
succeeding is the same throughout, is termed an Arithmetical progres
sion, for the summation of which, a general formula with its investiga
tion is given in Art. 143 of the “ Treatise on Arithmetic and Algebra”
published by the Society. The formula there is
n (2<s + (n — 1) b)
2