Single Premiums and Annuity Values   45

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5. Also Ax = l [vdx+v2dx+1 + ... +vndx+n_ + .... ]

or Ax=vax—ax=(1—d)ax—ax=l—dax or 1=dax+Ax.

The truth of each of these relations should be made evident in words.

+(vdx+v2dx+1+ . . . +vndx+n—1+ • • • )]

=1— dx [(1-v)dx+(1—v2)dx+1+ ... +(1—vn)dx+n—1+ . . . =1— d    [d+(1+v)d+1+...

lx

+ (1 +v +v2 + . . .+vn—1)dx+n—1 + . . . I

= 1 — d [lx +vlx+1 +v21x+2 + . . . +vnlx+n + . . ]

=1—dax,

or 1=dax+Ax• 7. By analogy,

1=iax+(l+i) Ax, so 1=7(,n)ax'n)+(1+-"))A(xm>, m

1=dax+Ax, so 1 =✓ (,,)a=m) +Axm),

6. Again, from first principles, we have

Ax=   [vdx+v2dx+1+ . . . +vndx+n_1+ . . . ]

= 1 [l_(d+d+1+... +dx+n_1+...)

lx