Let $a$ and $b$ be positive real numbers and $b \neq 1$. Find $a+2 b$ given that the exponential function $f(x)=a(b)^{x}$ passes through the points $P(0,3)$ and $Q\left(2, \frac{3}{16}\right)$. a) $\frac{3}{8}$ b) $\frac{7}{2}$ c) $\frac{3}{16}$ d) $\frac{7}{4}$ e) 4

Step 1 ：Given the points $P(0, 3)$ and $Q\left(2, \frac{3}{16}\right)$, we can create a system of equations using the function $f(x) = a(b)^x$:

Step 2 ：\[\begin{cases} a(b)^0 = 3 \\ a(b)^2 = \frac{3}{16} \end{cases}\]

Step 3 ：Solving the first equation, we get $a = 3$.

Step 4 ：Substitute $a = 3$ into the second equation: $3(b)^2 = \frac{3}{16}$.

Step 5 ：Solve for $b$: $b^2 = \frac{1}{16}$, so $b = \pm\frac{1}{4}$.

Step 6 ：Since $b \neq 1$, we choose $b = \frac{1}{4}$.

Step 7 ：Finally, find $a + 2b = 3 + 2\left(\frac{1}{4}\right) = \boxed{\frac{5}{2}}$