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Math Help - Mathematical Induction

  1. #1
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    Mathematical Induction

    Q: A farmer raises chickens in a farm. Taking into account various conditions, such as the sale of poultry, deaths, etc that affect the population of chickens in his farm, the number of chickens at the end of n months is modelled by Un, where

    Un = 0.875Un-1 + 50

    Express U1, U2 and U3 in terms of U0.

    Show that Un = (0.875^n)(U0 - 400) + 400 and deduce the value of Un when n becomes very large.

    Thank you in advance!!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Tangera View Post
    Q: A farmer raises chickens in a farm. Taking into account various conditions, such as the sale of poultry, deaths, etc that affect the population of chickens in his farm, the number of chickens at the end of n months is modelled by Un, where

    Un = 0.875Un-1 + 50

    Express U1, U2 and U3 in terms of U0.

    Show that Un = (0.875^n)(U0 - 400) + 400 and deduce the value of Un when n becomes very large.

    Thank you in advance!!
    Have you shown that this is true for a base case? n=0 will do.

    Now assume it true for some k, and consider:

    u_{k+1}=0.875 u_k+50,

    but by supposition: u_k=0.875^k(u_0-400)+400

    so:

    u_{k+1}=0.875[0.875^k (u_0-400)+400 ]+50=0.875^{k+1}(u_0-400)+350+50 =0.875^{k+1}(u_0-400)+400

    So from our assumption that the result was true for k we have proven it true for k+1, hence with the base case this proves the result of all n\ge 0

    RonL
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