1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Infinite geometric series

  1. Nov 26, 2015 #1
    1. The problem statement, all variables and given/known data
    hello this question is discussed in 2009 but it is closed now

    If you invest £1000 on the first day of each year, and interest is paid at 5% on
    your balance at the end of each year, how much money do you have after 25
    years?

    2. Relevant equations
    ## S_N=\sum_{n=0}^{N-1} Ar^n##

    where N is the last term
    r is the common ratio & A is a constant

    ## S_N= a\frac{1-r^N}{1-r} ##

    3. The attempt at a solution

    after 25 years i would set N=25 but this will give me a result of £47727. then i have to subtract £1000 because on the first day of each year i invest £1000 therefore i got a result of £46727

    but this is the wrong answer

    if i set N=26 i will get £51113 and then again subtract £1000
    therefore i got a result of £50113 and this is the right answer

    my question is why i must put N=26. isn't N the last term and equal to 25?
    and is this way of solution right?
     
  2. jcsd
  3. Nov 26, 2015 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Can't follow your equations. "N is the last term"? Wouldn't N be the number of terms/years ?
    And then after one year you don't have ##Ar^0## but ##Ar^1##.

    Tip: Don't change the notation from one line to the next. A is A, not a.

    ## S_N= A\frac{1-r^N}{1-r} ## looks weird too. Don't you mean ##
    S_N= Ar\frac{\ 1+r^N}{1+r}## so that after 1 year you do have Ar ?

    --
     
  4. Nov 26, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No, his formula is correct as written: ##\sum_{n=0}^N A r^n = A(r^N -1)/(r-1)## is a standard elementary algebra result. Of course, it also equals ##A(1-r^N)/(1-r)##.
     
  5. Nov 26, 2015 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

    If you sum from 0 to N, it is N+1 elements. The correct formula is ##\sum_{n=0}^{N-1} A r^n = A(r^N -1)/(r-1)##
     
    Last edited: Nov 27, 2015
  6. Nov 27, 2015 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Indeed: that was a typo on my part. I had intended to make the upper summation limit equal N-1, but somehow slipped up.
     
  7. Nov 27, 2015 #6

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Sorry for the minus sign - weak moment.

    Time to define what A stands for. My impression was that after one year the guy has ##Ar##, not ##
    \sum_{n=0}^{0} A r^n = A(r^1 -1)/(r-1) = A##

    Time to define what N stands for, too: at the begining of year 2, they guy has
    ##
    \sum_{n=0}^{1} A r^n = A(r^2 -1)/(r-1) = A(r+1)## and he invests another A ?

    I figured r = 1.05 and A = 1000 pounds. Where did I go all wrong ?

    $$1000 \,{1.05^{26} - 1\over 1.05 -1} = 51113.45 $$
    $$1050 \,{1.05^{25} - 1\over 1.05 -1} = 50113.45 $$ which OP considered the right answer .

    The
    in the template is clearly useful. And includes the list of what values in the problem statement the symbols used stand for ...

    :smile:
     
    Last edited: Nov 27, 2015
  8. Nov 27, 2015 #7
    yes you are right N is the number of terms
    and i get it now. first i invest £1000 so i have nothing at this moment after one year i have £1000*1.05= £1050
    so i start with £1050=Ar
     
  9. Nov 27, 2015 #8

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Easier: just write it out correctly from the start. If ##r = 1.05##, ##A = 1000## and ##N = 25##, the value you want is the "future value" ##F##:
    [tex] F = A r + A r^2 + \cdots + A r^N = A r \sum_{n=0}^{N-1} r^n [/tex]
    Basically, the final ##A## at the start of year 25 becomes ##Ar## at the end of year 25. The second-last ##A## at the start of year 24 becomes ##A r^2## at the end of year 25, and so forth.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Infinite geometric series
  1. Geometric series (Replies: 4)

  2. Infinite series (Replies: 2)

  3. Geometric series (Replies: 3)

Loading...