ANNUITIES. 211 



2. Carry the series to as many terms as the nunoiber of 

 years, and find its sum. 



3. Multiply the sum thus found by the given annuity, 

 and the product will be jthe amount sought. 



EXAMPLES. 



less upon every succeeding year to that preceding the last, which 

 has but one year's interest, and the last bears no interest. 



Letr, therefore, = rate, or amount of il. for i year ; then the series 

 of amounts of il. annuity, for several years, from the first to the last, 

 is I, r, r*, r', &c. to r^-'. And the sum of this, according to 



fj I 



xhe rule in geometrical progression, will be , = amotmt of 



•ll. annuity for / years. And all annuities arc proportional to 



their amouots, therefore i : - — - : : si : -^~. x » = 



r — I r — I 



amount of any given annuity «. Q^ E. D. 



Let r = rate, or amount of il. for one year, and the other 



letters as before, then y(nz=a, and =zn, 



r — I r- — 1 



And from these equations all the cases relating to annuities, or 

 pensions in arrears, may be conveniently exhibited in logarithmic 

 terms, thus : •> 



I. Log. n-\-Log» r^ — I — Log. r — i -zzLog, a* 



II. Log. a — Log. r^ — i -\-Log. r — izzLog. n. 



III. itS-^r-a^-n-Lo!..,,^^^ I V. r'- 21 + f- - I = o. 



