450 New Theorems respecling the Value of Annuities. 



And the analytical expression for the sum of each series will 

 be respectively : 



•"■ n 



y__ .r-l.(l + r-l)_n 



ix-1) 

 x+ 1 



Y = " ' 



ili_ n+l 



_ .r— I.(l+r-l)-(2'?+l +^77 + 1?) 



6v 



7? + 1 n+ 1 



n >r-l.(l +r- l) + r.l +r— 7-.i;;+ 13+ 3(r.H+ H + '?„^2 + r.7(+ 1)— r 



As an example in each case when the term is five years, an(i 

 rate of interest 5 per cent., we have for 



X 16,03823025 



Y 57,56850625 

 Z 232,4440064, 



But which numbers, by the erroneous process of computation 

 given in the work already quoted, would come out successively 

 for this term and rc^te of interest: 



X 17,11553125 



Y 64,03215625 

 Z 265,19378125. 



The difference in each case, between the amounts or the ex-i 

 cess in error, rapidly augmenting as the rate of interest and terni 

 of years increase. 



To the fornrjulse enumerated above, may be added the follow-: 

 jng : 



_JL_ n 7,-1 



(x-l) ' 



And this is a general expression for the amount of an an-: 

 nuitv increasing according to the numbers I. 3. 5. 7 . . •2n — 1. 



And if the annuity is supposed increasing in the order of 

 their squares, the general expression for its amount in n years 

 ■will be 



8.r 

 n n 



X--1 (1 +r- l) + r,l ^r-r (9ii + ]--)—Sn 



I intend at a future opportunity giving the necessj^ry theorems 

 for the present value of annuities under the particular circum- 

 stances just stated, and also some other matter relative to the 

 determination of the rate of interest in annuities. 



5, Haberdashers Place, Hoxtoi), JaS. B?NJ. BeNWBLL. 



Dec. 10, 181C. 



^cm. Oil 



