ANNUITIES. 207 



ANNUITIES. 



An Annuity. is a sum 01 money payable every year, for 

 a certain number of years, ox for ever. 



When the debtor keeps the annuity in his own hands, 

 beyond the time of payment, if is said to be in arrears. 



The, suni of all the annuities for 'the time they have 

 been forborn, together with the interest due upon each, is 

 called the amount. 



If an annuity be to be bought ofF, or paid all at once, 

 at the beginning of the first year, the price, which ought 

 to be given for it, is called t\\Q^ present nvorth. 



21? Jind the Amount of an Annuity at Simple Interest. 



RtJLE.* 



I. Find the sum of the natural series of numbers i, 2, 

 3, &c. to the number of years less one. 



2. Multiply 



* Demonstration. Whatever, the tipie is, there Is due up- 

 on the first year's annuity, as many years* interest aS the whole 

 number of years less one ; and gradually one less upon every suc- 

 ceeding year to the last but one ; upon which there is due only 

 one year's interest, and none upon the last ; therefore in the 

 whole there is due as. many years' interest of the annuity, as the 

 sum of the series i, 2, 3, 4, &c. to the number -of years less 

 one. Consequently one yearns interest, multiplied' by this sum, 

 must be the whole Interest due ; to which If all the annuities be 

 added, the sum is plainly the amount. Q^ E. D. 



Let r be the ratio, n the annuity, t the time, and a the amount. 



Then will the folio v/ing theorems give the solutions of all the 

 dliTerent cases. 



T t'^rn — trn . ^_ la — 2/« 



1. \-tnzza. II 



III. 



