Call  P^  the  present  value  of  P  dollars  due  in  71  years,  r
being  the  annual  rate  of  interest.      Then

P

P(iJ^ry  =  P,      whence     P,  =  , — , — r-.      .      (14)

Let  P.,  represent  the  sum  of  money  which,  at  the  rate  r,
will  produce  P  dollars  at  the  end  of  every  period  of  //  years,
in  perpetuity.
COMPARATIVE   COST  OF  CONSTRUCTION:  121

We  must  have     P.,{\  -{- r)"  =  P -\- P^,     or

whence

•       •  ^^  =  (i+^)»-i (^5)

An    annual    payment  of  P  dollars  running  for  n  years  at
the  rate  of  interest  r:
The  first  payment  amounts  at  the  end  of  n  years  to  P{i  -|-  r)" ;

second        "  "  "  "  "         "         /^(l   -j-  ;')«-i;

third         "  "  "  "  "      "  Pi^ I -f ;-)"--%  etc.
The  last  payment,      "  ''  "  '*      "    /'(i  _|_  r).
The  total  5  =  P\{i  +  r)"  +  (i  -f  r)— -f  etc.  +  (i  +r)|.
