Annuities   39

1. an ultimate 4, the probability of the payment of which is a plus an ultimate ;, the probability of the payment of which is 4 plus an ultimate 4, the probability of the payment of which is 4, the total ultimate value being 4X*+4X}+4X -=a.

2. the interest during each year of life which is

4 of a year's interest on 4 or a year's interest on

of a year's interest on 4 or a year's interest one 4 of a year's interest on 4 or a year's interest on r-1,

in all, a year's interest on 8.

Therefore ax4) =ax+8, approximately.

Similarly it may be shown that,

ax(m) ax+ 2m1 , approximately.

If m be infinitely great, we obtain the value of an annuity of 1 per annum payable momently

dx=ax+z, approximately.

Strictly, dx=J v px.dt=J dal apx

o   o

co

10. To find the value of a(''!) or Inan(m), we have

axn) = I n axm) =al") -71n• npx' aL+n 1 m—1   m—1

= ax+—   — npx (x+n+--), approxi-

2m   2m   mately

=ax, + 2m1 (1 —vnnpx)

(m)

I   =n

n ax   y npx ax+n+ m   J

2—m1/

(m) =vnnpx X rax+ (m)   n   (m)

n I rax   n =v npx ax+n:~

[ax+n.   m—1

=Vnnpx   r , + 2m    (1—tirrpx+n)

also and