Q1341 A lazy weather forecaster predicts that future maximum temperatures will be the average of the preceding two days maximum temperatures. The forecaster starts his forecast by noting that yesterday's maximum temperature was $23^\circ$C and the day before it was $29^\circ$C. What temperature in $^\circ$C is the weather forecaster's long term maximum temperature forecast?

ANS: The forecast temperature $T(n)$ on day $n$ satisfies the recurrence relation $$T(n)=\frac{1}{2}T(n-1)+\frac{1}{2}T(n-2)$$ with initial conditions $T(1)=29, \ T(2)=23$.

 

If we seek a trial solution of the recurrence relation in the form, $T(n)=a\lambda^n$ then we find this is a valid solution for any $a$ provided that $\lambda$ is a solution of the quadratic equation $\lambda^2-\frac{1}{2}\lambda-\frac{1}{2}=0$. The solution of the quadratic equation yields $\lambda=1$ and $\lambda=-\frac{1}{2}$ thus $$T(n)=a+b\left(-\frac{1}{2}\right)^n.$$

If we use the initial conditions $T(1)=29, \ T(2)=23$ then we find that $a$ and $b$ must satisfy the simulatenous equations

 

$$a-\frac{b}{2}=29$$

$$a+\frac{b}{4}=23.$$

 

We then solve for $a=25$ and $b=-8$ so that

$T(n)=25-8(-\frac{1}{2})^n$ and for large $n$ we find the

long term forecast $T=25 ^\circ$C.