# Mathematical Induction

• May 12th 2008, 05:12 AM
Tangera
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.

• May 12th 2008, 01:25 PM
CaptainBlack
Quote:

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? \$\displaystyle n=0\$ will do.

Now assume it true for some \$\displaystyle k\$, and consider:

\$\displaystyle u_{k+1}=0.875 u_k+50,\$

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

so:

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

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

RonL