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