An  annuity  may  be.  (1)  terminable,  L  e.  payable  only  for
a  certain  number  of  years,  (2)  perpetual,  L  e.  payable  for  ever,
(3)  deferred,  L  e.  payable  regularly  after  a  certain  number  of  years
have  elapsed.
Insurance  premiums,  rent,  pensions,  &c.,  are  common  examples
of  annuities.
266.  If  £1  is  allowed  to  accumulate  at  compound  interest  at
t*  per  cent,  for  n  years,  it  amounts  to  ( 1  +  r-^  j  •

1  +  r^  is  usually  denoted  by  i?.
Hence  in  n  years  at  r  per  cent.  P  pounds  amounts  to  PA**.
If  P  denotes  the  money  invested  now,  and  A  the  sum  to  which
it  amounts  at  the  end  of  n  years,

A  =  PR",

It  follows  from  this  formula  that  the  possession  of  £P  now  is
equivalent  to  the  possession  of  £A  in  n  yean*  time,  i.  e.  £P  is  the
Present  Worth  of  £A  due  n  years  hence.
The  formulae  for  a  geometric  progression,  together  with  the
formula  A  «=  Pfi^,  are  all  that  are  required  to  deal  with  elementary
questions  on  Annuities ;  no  special  formulae  are  needed.
267.  Example  I.  The  rent  of  a  house  is  £45  per  annum,
payable  yearly.  What  would  he  a  fair  price  to  pay  for  a  21  years'
lease  of  the  house,  interest  being  recleoned  at  4  per  cent,  9


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400  GEOMETRIC  PROGRESSION

Instead  of  paying  j646  at  the  end  of  each  of  21  years,  the  parelia<
makes  a  single  payment  at  the  beginning  of  the  21  years.
A ''PR!*.
r  "  4,       .-.   12  -i  1-04  and  ^  »  45.
