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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 ?

    How have you started to attack the problem? The objective here should be quite clear, find the sum in both cases.

    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?

    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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