l<-c(10000000,9795800,9782674,9770739,9759796,9749646,9740091,9731033,9722372,9713914,9705657,9697407,9689164,9680831,9672409,9663801,9655007,9646028,9636864,9627516,9617888,9607982,9597798,9587240,9576311,9565011,
     9553342,9541209,9528519,9515274,9501382,9486845,9471571,9455469,9438544,9420611,9401676,9381556,9360166,9337421,9313144,9287253,9259577,9229946,9198195,9164070,9127414,9088075,9045725,9000135,8950994,8898004,8840879,
     8779258,8712711,8640918,8563495,8480001,8389943,8292787,8188132,8075463,7954250,7823959,7684067,7534074,7373448,7201720,7018508,6823464,6616235,6396708,6164763,5920515,5664157,5396186,5117257,4828285,4530476,4225258,
     3914448,3600118,3284604,2970563,2660793,2358287,2066119,1787317,1524778,1281103,1058511,858696,682723,530974,403084,297988,213982,139088,73021,23732)
f<-function(i,x,k){
  v<-1/(1+i)
  D<-c()
  for(j in 1:100){
    D[j]<-v^(j-1)*l[j]
  }
  N<-c()
  N[100]<-D[100]
  for (j in 1:99){
    N[100-j]<-N[100-j+1]+D[100-j]
  }
  V<-1-(N[x+k]-N[x+10])*D[x]/((N[x]-N[x+10])*D[x+k])
  return(V)
}

P<-function(i,x,k){
  Ps<-f(i,x,k+1)/(1+i)-f(i,x,k)
  Pk<-(1-f(i,x,k+1))*(l[x+k]-l[x+k+1])/(l[x+k]*(1+i))
  return(c(Ps,Pk))
}