[ 449 ] 



XCII. New Theorems applicable to the Value of Annuities. 



To Mr. Tillbch. 



Sir, — X HE following correct and new theorems, for deter- 

 mining in all cases the amounts of annuities certain when in- 

 creasing in the constant ratio of the naturahiumbers 1.2.3. . . .w, 

 and also of the squares and cubes of the natural numbers, may 

 prove acceptable to many of your readers. 



The method of summing of series of this kind is very well 

 known, hut in this instance 1 was led to them by a process es- 

 sentially different. However, my object is not to discuss methods 

 and principles, but to detect error with a view to the consecjuent 

 establishment of truth ; and as this matter appears not to have 

 been clearly illustrated by any of the writers on series or an-" 

 miities with whom I am acquainted, 1 am therefore desirous to 

 announce these theorems in your publication. Mr. Baily, in his 

 Doctrine of Annuities, has given the investigations relating to 

 this subject ; but in the several series there made to represent, 

 respectively, the amount of these annuities, a remarkable incon- 

 sistency exists, the consequence of which is, that the f >rinulae, 

 thence derived, are of no avail in satisfying the oljjects of our 

 inquiry when the annuities are thus annually increasing ; and 

 further, that thev are applicable to this purpose, only on the as- 

 sumption of the annuities being actually decreasing in the order 

 specified aljove. It is therefore proper that the two assumptions 

 be separately distinguished, and this auibiguity in the formulae 

 pointed out: otherwise the adoption of it indiscriminately,, may 

 not only confuse and mislead, but produce egregious mistakes. 

 At the same time, I cannot but express my surprise that an au- 

 thor, who censures with so much freedom and severity the scien- 

 tific labours of his contemporaries, sliould have suffered such 

 absurdities to deteriorate vvhat in every other respect is a very 

 useful and popular performance. 



I shall now proceed to give the theorems themselves : in order 

 to which I take X to denote the amount of an annuity increasing 

 according to the order of the naturul numl)ers \. 2.'^. . . .n; 

 and Y and Z that inereasins?, by the squares and culies of them. 



Now putting X to denote 1 -f r the amount of 1/. for a year, 

 and n the number of years ; then these several (juantities will be 

 ccjuai to and truly represented l)y the following different series: 



X = « + (n— l)j-f (n — 2;j:*+(w — 3)x''+(h — 4>'*+(/; — 5)j;* a:""' 



y = n'+(«-l^j+(n— 'i->-+(n-3')x +(n— 4«}x'+(n-5''):t" ar''"^ 



Z=n'+(«-l')r+(«-2^)f-+(«-3^)r' + («-4^)x^ + (n-5^)x'' x""' 



Vol. 48. No. 221. Dec. IblG. F f And 



