Assurances, Single Premiums   33

The true value of Ax is I v`,pxµx+:. dt.

0 n

Similarly Ax n = v`gpxllx+t. dt.

0

(2) We have A=Ax(2)—vnnpxAx+n

x

= (1+i)Ax—vnnpx(1+4i)Ax+n = (1+ i)Axn

(ml A _

X(1+m-

and   2m z)Axn'

But   Axnl — (m)_Azn (m)

+Axn

1 n a — 1 i f-    2m ) A ' -

~ +A , x   .

Ix

This may be regarded as the arithmetric mean of lx quantities, namely, dx each of the value v, dx+1 each of the value v2, etc.

lx+lx+1+lx+2+ ... .

Now, vl+ex =v   lx

dx+2dx+1+3dx+2+ ... .

=v   lx

_ [dx (v2)dx+l (v3)dx+2   !x

which also may be regarded as the geometric mean of the same lx quantities.

Now the A.M. of any number of positive quantities, not all equal to each other, is greater than the G.M. of these quantities, therefore Ax>vl+ex,

10. Consider Ax= vdx+v2dx+1+v3dx+2+ • • • • • •