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Thread: 1967 HSC help

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    1967 HSC help

    This is only the first question of the 1967 Extension 2 HSC and I am already having trouble. Pls help. Are there solutions to these older papers (<1991) online??

    Screen Shot 2017-08-05 at 12.39.26 pm.png

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    Re: 1967 HSC help

    You could use mathematical induction.

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    Taking a break! dan964's Avatar
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    Re: 1967 HSC help

    i will check if I have anything.
    pretty sure though that everything I have is up.
    2014 HSC
    2015 Gap Year
    2016-2018 UOW Bachelor of Maths/Maths Advanced.

    Links:
    View English Notes (no download). Purchase a downloadable/printable copy on SELLFY here + Year 10 notes
    Maths Notes - Free Sample
    Chemistry Notes Sample + Buy (soon)

    thsconline: https://thsconline.github.io/s/ (2017 papers uploaded)

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    Re: 1967 HSC help

    Consider:
    m.a^m = a^m + a^m + ... + a^m (m terms)

    m.a^m > a^0 + a^1 + ... + a^(m-1) (since a>0 and m is a positive integer)

    m.a^m + [m.a^(m-1) + ... + m] > a^0 + a^1 + ... + a^(m-1) + [m.a^(m-1) + ... + m]

    m.a^m + m.a^(m-1) + ... + m > 1 + a^1 + ... + a^(m-1) + m[a^(m-1) + ... + 1]

    m.a^m + m.a^(m-1) + ... + m > 1 + a^1 + ... + a^(m-1) + m[a^(m-1) + ... + 1]

    m(a^m + a^(m-1) + ... + 1) > (m+1)(1 + a^1 + ... + a^(m-1))

    (a^m + a^(m-1) + ... + 1)/(m+1) > (1 + a^1 + ... + a^(m-1))/(m)

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    Re: 1967 HSC help

    The left hand side is the average of (m+1) numbers {a^0,a^1,a^2,...,a^(m)} and the right hand side is the average of m numbers {a^0,...,a^(m-1)}. Since a^m is bigger than all the other numbers, having that present will increase the average. You can instead prove that adding a number greater than the average to any set of (n) numbers will increase the average of the (n+1) numbers.

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