84   Life Contingencies

15. Show that

1 +ex = 4x +px (1 +4x+1) +2Px (1 +4x+2) +etc.

16. If µx =   1    , find lx, and the complete expectation of life.
100 —x

17. Obtain an expression for the average age at death of the present members aged x and upwards in a stationary community supported by births alone and unaffected by migration.

If the law of mortality is as indicated by the following values of dx—the limiting age of the table being 100   find the average
age at death of the present members aged 94 and upwards.

d94-11, d95=9, d96=7 and so on.

18. What do the following integrals represent? rn

0 n+m

2.   gPx • tax+t . dt

n

3. f (gyp),+IA—tpyz) • tt'x • tax+a . dt.

0

19. Given A25=.356, p26=.992, and i = .03, find A26.

~n

20. Prove Ax =i   d,I . vzpx—d v a [(I+ .\ (1+2)n—1

   T.   ,—fps]+ 1.

n=1   n=1

21. Give a formula for the single premium at age x for a whole life assurance, the sum assured during the first year to be (1.015), during the second year (1.015)2, during the third year (1.015)3 and so on.

x+l

22. Prove J (µx+S)dx=colog vpx. x

23. Prove Axxx=Axxx-3(Axx-Ax).