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XLIV. New Theorems Jfor determining the Rate of Interest, 

 and the Value of increasing Annuities, fefc. Is'c. 



To Mr. Tilloch. 



Sir, — xjLs you were pleased on a recent occasion to give a 

 favourable reception to a paper of mine on the subject of in- 

 creasing annuities, I am induced to communicate the present 

 one, which may in a degree be considered as a continuation of 

 the former. I shall first, however, give the necessarv theorems 

 for the amount and present value of an annuity increasing ac- 

 cording to the numbers 1.3. 6. 10. 15, or figurative numbers of 

 the first order; and the succeeding two will be general expressions 

 •for the sums of the series, denoting the present value of an- 

 nuities increasing in the manner stated in the latter part of the 

 paper already referred to. 



First, then, of an annuity increasing annually, by the numbers 

 1.3.6. 10.. , ""'"- the general theorem for its amount in n 



years, putting x for \-\-r. the amount of \l. in a year, will be 



2x n + I 



2(.r— 1)'! 



And that for the present value of the same annuity is 



2x -~n —n — 11 



^_, (.T — x)— {"in + 1 ..r + JT— 1 (?i- + ?,n + 2).iJ^ 

 ' 2(jr-lji 



Again, the expression for the present value of an annuity in- 

 creasing in the ratio of the numbers 1. 3. 5. 7. 9. 1 1 is 



(x+\) — {2. x+ Cr— 1.2n+ l..r)) _ 

 {x-\)'^ ' 



and if it be siipposed increasing by the squares of these numbers, 

 the present value will be expressed by 



8 —n — n —n 



'j^ZTx ix~x)+x—\ -8(n + 1 T)-x—\.x.'in+ l^) 



'(x-l)'- 



In each of the three last theorems, if n is infinite, or the an- 

 nuity is a perpetuity ; thei/, as all the quantities affected by n 

 vanish, or become nothing, they will be reduced to these fol- 

 iowitig : 



\{x—i)i/ V(.r-l)V \(.r-l)3/ 



Which therefore are general expressions for the value of the 

 perpetuities under the above circumstances respectively. 



M 2 As 



