1. ## 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.

2. Originally Posted by Tangera
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.

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