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    sarah gets pocket money every week from the age of 5 until her 21st birthday and is given a choice of 2 options options . option 1:get the same number of pounds each week as her age. option 2 : get 5 pounds a week aged 5 and increasing by 15% a year. which option should Sara choose ?
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    How have you started to attack the problem? The objective here should be quite clear, find the sum in both cases.
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    You have two sums to calculate, one arithmetical and one geometric.Your initial term is the money received per year, with either an increment per year, or a factor per year. What are those numbers? How will you calculate the sums?
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    Whilst you can memorise the equations for the sums of these series, they're quite quick to derive.

    Here's the derivations for the sum of a geometric progression:
    S = a + ar + .. + ar^n-1
    rS =    ar + .. + ar^n-1 + ar^n (1-r)
    \therefore S-rS=a-ar^n=a(1-r^n)
    \therefore S = a\frac{(1-r^n)}{1-r}

    And for the sum of an arithmetic series:
    S = a + (a+d) + .. (a+(n-1)d)
    S = (a+(n-1)d) + (a+(n-2)d) + .. a
    \therefore 2S = n[2a+(n-1)d]
    \therefore S = \frac{n}{2}[2a+(n-1)d]
 
 
 
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