Examples   85

24. Show that

A.yyz =As, z =Ax+Ay,z —Ax= Az +A y } Az —Ax,, A,,z —Axs +Axyz

25. Examine the following statements critically:

1.  1=iaxn,+(1+i)Axn

2. ax —axn = vnax

3. 1 =daxrzi +Agra' +v;;px• Explain any corrections that may be necessary.

26. Explain (i) Ax+iAx —qx(1—Ax+1) =Ax+1 (ii) ax+iax—px(1+ax+1)=0.

27. Show that   [v" ax= E   (1+i) (1+i)n—1 (n-lt'x—npx)].

   n=1   Z

28. Prove that az2,1 = 4 (3axni + axn1 ) approximately.

29. Show that (i) axy : zw = ax xw + ay zw — axyzw

(ii) axyzw = axy zw = axy + asw — axy zw =lax — axy + r■axyz — axyzw.

30. Find the H'r. 3% value of

1. An annuity payable quarterly for not more than ten years to a life now aged 60. What correction would you suggest to make the annuity apportionable?

2. An annuity payable monthly, first payment at age 70, to a life now aged 50. The rent to be $100 a month from age 70 to age 80 and $125 a month thereafter.

31. The value at 3% of an apportionable annuity payable yearly is 12.100. Find the value of an annuity on the same life payable quarterly but not apportionable.