Question 52·Hard·Advanced Math

The same $c$ and $r$ must work at every input. Replace $f(2)$ using the rule: $cr^2=32$. Replace $f(1)$ and $f(3)$ using that same rule:

Type the two equations as one list regression, using $\sim$ to tell Desmos to fit the constants. Add the restrictions because the problem says $c$ is positive and $r>1$. Under PARAMETERS, Desmos reports $c=2$ and $r=4$.

Define $f(x)=cr^x$ below the regression, then type $f(0)$ and $f(4)$. Desmos uses the fitted constants and prints $f(0)=2$ and $f(4)=512$. The zero exponent makes $f(0)$ equal to $c$; it doesn’t make it $0$.

Type $f(0)+f(4)$; Desmos prints $514$. That adds $2$ and $512$ to give the requested sum, rather than stopping at $f(4)$. So $f(0)+f(4)=514$. Choice C.

Each increase of $1$ in the input multiplies an output by $r$. Step backward from $f(2)=32$ by dividing by $r$: $f(1)=\frac{32}{r}$. Step forward by multiplying by $r$: $f(3)=32r$. These are the two outputs whose sum is given.

Substitute those outputs into their given sum: $136=32\left(\frac1r+r\right)$. Divide by $32$: $\frac{136}{32}=\frac1r+r$. Reduce the fraction: $\frac{17}{4}=\frac1r+r$. This finds the sum of the factors without finding either constant.

The target inputs are two steps from $2$, so they use $1/r^2$ and $r^2$. Square the factor sum: $\left(\frac1r+r\right)^2=\left(\frac{17}{4}\right)^2$. Expand; the middle term is $2$ because $r(1/r)=1$: $\frac1{r^2}+2+r^2=\left(\frac{17}{4}\right)^2$. Subtract $2$:

Multiply those two-step factors by $f(2)=32$ to get the requested sum: $f(0)+f(4)=32\left(\frac1{r^2}+r^2\right)$. Substitute the expression from the previous step: $f(0)+f(4)=32\left[\left(\frac{17}{4}\right)^2-2\right]$. Type that last expression in Desmos. It prints $514$, so $f(0)+f(4)=514$. Choice C.