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    At the beginning of the year 2000 a company bought a new machine for £15 000. Each
    year the value of the machine decreases by 20% of its value at the start of the year.

    (a) Show that at the start of the year 2002, the value of the machine was £9600.


    When the value of the machine falls below £500, the company will replace it.
    (b) Find the year in which the machine will be replaced.
    To plan for a replacement machine, the company pays £1000 at the start of each year into a savings account. The account pays interest at a fixed rate of 5% per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.

    ------

    I got the right answer to a but i'm struggling to understand an aspect for b.

    my working for b:
    ar^n < 500
    15000*0.8^n < 500
    0.8^n < 1/30
    nlog0.8 < log(1/30)
    n > 15.24

    From this, I would have said that the year it was replaced was 2016 because n has to be a whole number and whenever I put 15 as n, it gives a value higher than 500. But the answer in my book says it is 2015 but I dont understand why.
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    (Original post by blankboi)
    At the beginning of the year 2000 a company bought a new machine for £15 000. Each
    year the value of the machine decreases by 20% of its value at the start of the year.

    (a) Show that at the start of the year 2002, the value of the machine was £9600.


    When the value of the machine falls below £500, the company will replace it.
    (b) Find the year in which the machine will be replaced.
    To plan for a replacement machine, the company pays £1000 at the start of each year into a savings account. The account pays interest at a fixed rate of 5% per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.

    ------

    I got the right answer to a but i'm struggling to understand an aspect for b.

    my working for b:
    ar^n < 500
    15000*0.8^n < 500
    0.8^n < 1/30
    nlog0.8 < log(1/30)
    n > 15.24

    From this, I would have said that the year it was replaced was 2016 because n has to be a whole number and whenever I put 15 as n, it gives a value higher than 500. But the answer in my book says it is 2015 but I dont understand why.
    The n^{\text{th}} term of a geometric sequence first term a and ratio r is ar^{n-1}.

    Not ar^n.
 
 
 
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