. e^^V>*i»<.^. - 071 Successive Lives. ^''' ^' '"'* '' 11 



= I 1 J. _ Ifl — nM— I 



«.= (i+7)(a'-a'a") 



«3= (l + iVa'a"-~a'a"a'") 



«„= (l + iVa'a"...a('»-»)-a'a"...aW). 



Hence by addition we have 



A«= (l+^)(l-a'a"...aW)-l . .... (10.) 



It appears from (4.) that (10.) is equivalent to 



A„= (l+-)(l-a„)-l; 



and, in fact, this might have been deduced immediately from 

 (3.), for the formula (1.) which expresses the relation between 

 an annuity and the corresponding assurance, is true of any 

 status*, providing the status be such that the assurance on it 

 must be paid some time or other. 



Problem IV. A copyhold estate is held on a certain num- 

 ber of lives A, B, C . . . . and each life is renewable at the end 

 of the year in which it may fail by paying a fine of / pounds. 

 Required the present value of all the fines. ^ 



Let A', A" .... be the lives that succeed A ; B', B" ...', 

 those that succeed B, &c. ; also let a, a', a" .... be the values 



of assurances on the lives A, A', A" ; b, b', b" . . . . those 



on B, B', B" . . . ., &c. 



By (4-.) it appears that the fines (each equal to/*) payable 

 at the deaths of A, A', A" . . . . are now worth /a, /aa', /aa'a", 

 .... respectively ; and similar expressions being true of the 

 other successive lives, the present value of all the fines is, 

 f a + aa' + aa'a" + ..... 



+b+bb'+bb'b';+ ^ . _ . (11.) 



•' 1 +c+cc'+ cc'c" + ... ^ ^ ' 



■+ 



!•■•■ 



' By the status of an annuity, I mean the state or condition of things 

 during the continuance of which the annuity is to be paid." — De Morgan's 

 Essay on Probabilities, ^.\2Q. . ■- 



