60   Life Contingencies

Writing   CS for vx+ldx, Dx for vxlx

and   Mx for Cx+Cx+1+Cx+2 + . . . to end of table,

we have Ax =   x , for all values of x. The whole series of values

x

of Ax for any mortality table at any rate of interest can be found at once after the columns Mx and Dx have been formed for that table and at that rate of interest.

3. Similarly for annuity values

ax = 11 [vlx+1 +v2lx+2 1 'v3lx+3 + • • • ] x

L [vx+11x+1 +vx+21x+2 +vx+31x+3 +•

v   •   ]

x

1

= D [Dx+l +Dx+2 +Dx+3 + • • ]

Dx

and writing Nx for Dx+1+Dx+2+Dx+3+ or   Nx for Dx+Dx+1+Dx+2+ . . .

we have   ax = Nx , and ax — = Nx
—.

Dx   Dx

The difference between Nx and Nx must be noted Nx =Dx+Nx =Nx-1.

Also since   dx =lx -1x+1

vx+1dx =vx+1lx —vx+1

1x+1

therefore   Cx =vDx —Dx+1

and      Mx =vNx_1 —Nx
= vNx — Nx+1 .

4. The following numerical illustration of the deduction of values for ax from known values of lx and vx may be helpful.