Commutation Symbols   63

D

but   Axnl =vnnpx=    Dx±n•

Therefore Axe =AX n1 +Axe = AIxMx+n+Dx+n Dx

n lax = Nx+n = Nx+n+l = Dx+n ax+n Dx   Dx   Dx

Nx —Nx+n Nx+1 —Nx+n+1

axis =   _

Dx   Dx

Nx+n Nx+n—1 Dx+n

n l ax =    =   =    a.x+n

Dx   Dx   Dx

and hence   axnl = Nx—Nx+n _ Nx-1 —Nx+n—1

Dx   Dx

10. For annual premiums, we have,

Px. ax =Ax so that P. Nx = Mx

Px axnl =Aznl so that Px,~I (Nx —Nx+n) = Mx — Mx+n Pxnl ax n =Axe so that Pxnl (Nx —Nx+n) =Dx+n

Pxnl • axttl =Ax~ so that Pxnl (Nx—Nx+n) = Mx —Mx+n+Dx+n and nPx axn, =Ax so that nPx (Nx —Nx+n) =Mx.

11. The commutation symbols also enable us to obtain workable expressions for increasing or decreasing assurances and increasing or decreasing annuities.

(IA )x = I [vdx+2v2dx+1+3v3dx+2+4v4dx+a+ ... ]

1

= D [Cx+2Cx+1+3Cx+2+4Cx+a+ ... ] Dx

= -[Mx+Mx+1+Mx+2+ . . . ] Rx

9. Also,

Dx