INTEREST. 



The value of i/., pavable at the end of n years, being m _ , j j. "» r «. 



■^ - , — therefore vnll be = i -t- r. , and /> = - — i • Ji irom 



=:;,, its value at the end of n — i years will be— — ^^_, *- 



J + ri- I +11' . r 



at the end of n — 2 years = ^. , and fo on. It fol- which r will be found = — -— • 



,^-,-1— mp 



lows, therefore, that the prefcnt value of the annual payment „ c \ ^. m 



' ^ r t ■> ^ Q^j^ , ^^ ^^ p bcmg given to find r. Since - =■ 



of a for n years will be expreffed by the feries — -— f 



a _ a 1 + ;>, the log. I + r will' be 



' "" '' 1 + r, and confequently r, will be known. 



^===j„ = /; which, therefore, is given when a, n, and Ca/f j.-n., r, /. being given to find n. Tliis is derived 



»• are given. If/, «, r are given, the value of a will be ob- from the preceding cafe, being = — ===^-. 



tained, being = Vi^^—^ ' . If a, r, p are given, the value q^j-^ 4.-0, m,p being given to find n . . . 1 + r'" being =^ 



log. a - log. a"^:^ -. ^, I -r H' - 1 "iU be known; and fince r is = 



of n will be obtained, beine = _s — ° i — In p 



log. I + r ^^ _ 



order to find the value of r, when a, p, n are given, let the Ij; '_!_ ^ ^ „i]i alfo be known, therefore, fince log,. 



I, a '" 



binomial 1 -t- /I" in the equation - - ,7;7^7i- = ■?' r+~^' is = n x log. FTT, n will be known, being = 



be expanded, &c., and we fliall have — = J , -. 



na ^ ' log. 1 -t- r 



n+i.n+2 ..„ r ,71 ,7+7 — f-''^/'' ^- — o, r, p being given to find m. 3v Ci^^f J, '«■ 



r + r , &c. ; confequently — ~ at 



^•3 ^^^^^^^^ "^ ^ may be calily found = -—f—. 



"^-T- ■HI n-l-I n+I.n+2,,, -;;t;_ a — pi 



— j«+' wiUbe= I ^'' + ^r '^'«'^^' — Trt/J 6— a, », / being given to find m. The value of r 



— J rnay be derived from one of the foregoing theorems, in which 



J 4 r ^^ — r' nearly. Let — j"^' — I = J, and a,/, n are given (being = ^ +•/ i4 — 2^1/), .and hence m = 



' ^ " / i p. r +^rJ" will alfo be given. 



*, then will r be = i -f- \,' bb — 2lrd. Cafe -j.—nt r, p being given to find m- As r is given, this 



» ~ ' value is immediately obtained from the preceding cafe, being 



Example I. — Snppofing a, n, r to be refpeflively equal to __ a _ fTITTl"; 



10, 21, and .DC, then will 6 be =z — — '° -. , = Cafe 8.— m, ;>, a being given to find p. By C^y^ i, r is = 



.05 .05.x 1.05I ■ ^;r^p,a ^ , am 



200 — 71.79 = I2S.2I. T — . therefore/- = — r-,,^-- 



Example 2.— Let ;;, r, and/> be refpeclive'ry equal to 21, ' " 



, p i_ ■„ , -o? X 105'" C/1/c- q.— 7;, »i, r being given to find *. Since — = "i + r'" 



.cj, and 12S.21, then will a. be = . / . 1— = y y > ' 6 fa /- ^ 



iTo5^'-' — I ra 



.1:0? X 12S.21 * will be = ^=,\ 



V-^ =10. ^ I + rl 



'■' Ca/* 10. — n, »i, a being given to find/. Let the value 



Example 3.— The quantities r and / being (liU the fame of r be found when m, n, a are given by one of the preceding 



ard.ij being equal 10, n will be = — T •5"' 5°34g _ jj,_ theorems (being = V ii -f- 2.ic — ^), and/ = "* 



.0211893 I + r'* 



Example \. — a,/, n being refpedively equal toio, 128,21, will then be given, 



and 21, r will be equal to V .06246 = .3 — .25 = Cnfe 11.— n, m, / being given to find a. Since — = 



.05. ,;;]■ 



The following twelve cafes, though they do not fo fre- ^ + '' "' '" "''" ^« = ;,'"" "' ^"'^ confequently given ; 



quently occur in pradtice as the preceding, may not im- , j. ^,. , r . r j r .u 



properly be added here. Let a, as before,, be the anauity, ^'^""^" '"^ ''^'^ ^'^'"^ "'^ " "=">: ^'^ ^""""^ ^'"'"^ '^^ ^^"^^'°" '" = 



u the number of years, m the amount of the annuity in n I + »' 

 years, / its value for the fame time, and r the rate 01 iii^ 

 tercft. 



Cafe I — m, p, a being given to find r. In the foregoing Cnfe 12. — r.i, p, r being given to find a. Let - be fub- 



•afes m beirg =: _: J , and/ = .fj-'^t '_^, Ilituted for its equal \ + ;' ' in the c.vprelTion " -| _ , 



a:id 



6 



