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    A competitor is running in a 25km race. For the first 15km, she runs at a steady rate of 12km/h. After completing 15km, she slows down and it is now observed that she takes 20% longer to complete each kilometre than she took to complete the previous kilometre.

    Find to the nearest minute the time the competitor takes to finish the race.

    I can't seem to do this. My method was finding the time taken to complete the first 15 kilometres (81.25 minutes), and adding this onto the time taken for the rest of the race, which is a geometric sequence. I can't seem to find the right answer though.

    My calculation for the time taken to complete kilometres 16-25 is shown below. The dotted line is supposed to represent the fraction line. I think this is where I'm going wrong. a = 5 ... this is the first term. This is calculated from 1/12. 1/12 hours = 5 minutes. 1.2 is the common ratio, as each KM takes 20% longer than the previous KM. There are 10 terms between 16 and 25, hence n=10.

    5 * (1 - 1.2^10)
    --------------------
    1 - 1.2


    When I add this to the time taken to complete the first 15km, I get a time taken of 3 hours 31 minutes. The correct time is 3 hours and 51 minutes. I have no idea where I went wrong :/

    Help would be appreciated, thank you!
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    (Original post by zebra015)
    ...
    So from 14km to 15km it takes 5 mins. Then from 15km to 16km it takes 5\cdot 1.2 mins. Then from 16km to 17km it takes 5 \cdot 1.2^2 mins, etc... etc...

    So you essentially have for the total total after the first 15km: 5\cdot 1.2 + 5\cdot 1.2^2 + ... + 5\cdot 1.2^{10}

    Your calculation for the sum implies that 5 is the first term of the sequence. It is not. The first term is 5\cdot 1.2.
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    Alternatively, you could calculate this as 14km @ 5min/km then an 11 term series with a=5, r=1.2 and n=11. Either way is perfectly OK but this way has the slight advantage that you don't have to calculate the value of "a" to plug into the series sum formula. But more important than minor variations in approach is to think carefully about what it is you're adding up.
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    (Original post by RDKGames)
    So from 14km to 15km it takes 5 mins. Then from 15km to 16km it takes 5\cdot 1.2 mins. Then from 16km to 17km it takes 5 \cdot 1.2^2 mins, etc... etc...

    So you essentially have for the total total after the first 15km: 5\cdot 1.2 + 5\cdot 1.2^2 + ... + 5\cdot 1.2^{10}

    Your calculation for the sum implies that 5 is the first term of the sequence. It is not. The first term is 5\cdot 1.2.
    Thanks a lot for the reply

    (5*1.2) * (1 - 1.2^10)
    --------------------
    1 - 1.2

    = 155.752

    155.752 + 81.25 = 3 hours 57 - still incorrect ... can't seem to see where I've gone wrong :/
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    (Original post by zebra015)
    Thanks a lot for the reply

    (5*1.2) * (1 - 1.2^10)
    --------------------
    1 - 1.2

    = 155.752

    155.752 + 81.25 = 3 hours 57 - still incorrect ... can't seem to see where I've gone wrong :/
    the first 15 km take 15*5 = 75 minutes, not 81.25
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    (Original post by zebra015)
    Thanks a lot for the reply

    (5*1.2) * (1 - 1.2^10)
    --------------------
    1 - 1.2

    = 155.752

    155.752 + 81.25 = 3 hours 57 - still incorrect ... can't seem to see where I've gone wrong :/
    I'm an absolute idiot
    Should be 155.752 + (15/12), which results with 3 hours and 51.
    Thank you to everyone that helped!
 
 
 
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