Advanced Math

According to the problem, the initial value is $A$, and it increases by $7.5$ percent each year. Hence, after one year, the value is

Similarly, after two years, the value is

Furthermore, after $x$ years, the value is

According to the problem, the value of the investment is $kA$ after $6$ years. Let’s solve for $k$.

We can see that

We know that if $f$ is a polynomial function and $f(a) = 0$, then $x - a$ must be a factor of $f(x)$. We can see from the table that $f(2) = 0$. Hence, $x - 2$ must be a factor of $f(x)$.

Subtract $1$ from both sides.

Add $1$ to both sides.

If $f(x) = 2^x$ and $g(x) = x + 2$, we get the function composition $f(g(x))$ by replacing $x$ with $g(x)$ in the expression of $f(x)$. Thus,

Similarly,

Therefore, the difference is

The equation

is a quadratic equation with $a = k$, $b = -k$, and $c = 2$. Let’s use the quadratic formula to find the solutions.

Thus, the sum of these values of $x$ is

Notice that the terms with square roots cancel each other out.