5. \( f(n)=10 \cdot 2^{n} \) at \( n=-2 \)

To find \( f(n) \) when \( n = -2 \), we simply plug in \(-2\) into the function: \[ f(-2) = 10 \cdot 2^{-2} \] Calculating it gives us: \[ f(-2) = 10 \cdot \frac{1}{2^2} = 10 \cdot \frac{1}{4} = \frac{10}{4} = 2.5 \] So, \( f(-2) = 2.5 \). To get a better grasp of exponential functions like this, remember that they can model growth or decay. For instance, in finance, they can represent compound interest, showcasing how money grows exponentially over time. With even a small interest rate, your pennies can turn into dollars! Isn’t that a fun twist on numbers? Exponential functions are also found in biology, representing population growth or decay, like how bacteria double in number under perfect conditions. Just be cautious with your bacterial friends; in a matter of time, they might be throwing a wild party in your petri dish!