If $2^{n-1} + 2^{n+1} = 320$, then $n$ is equal to

### Concept & Formula This question requires identifying common exponential bases and factoring them out to solve for an unknown exponent. It relies heavily on the **Multiplication Law of Indices**. The core principle used to break down the terms is: $$ a^{m+n} = a^m \times a^n $$ ### Step-by-Step Solution * **Step 1:** Express both terms on the left side in terms of the smaller power, $2^{n-1}$, to facilitate easy factoring. $2^{n+1}$ can be rewritten as $2^{(n-1) + 2}$, which is equal to $2^{n-1} \times 2^2$. The equation becomes: $2^{n-1} + (2^{n-1} \times 2^2) = 320$ * **Step 2:** Factor out the common $2^{n-1}$ term from both parts of the expression. $2^{n-1} (1 + 2^2) = 320$ * **Step 3:** Simplify the arithmetic inside the parentheses. $2^{n-1} (1 + 4) = 320$ $2^{n-1} (5) = 320$ * **Step 4:** Divide both sides by 5 to isolate the exponential term. $2^{n-1} = \frac{320}{5}$ $2^{n-1} = 64$ * **Step 5:** Convert $64$ into a base of $2$ to match the left side. $64 = 2^6$ So, $2^{n-1} = 2^6$ * **Step 6:** Since the bases are equal, equate the exponents and solve for $n$. $n - 1 = 6$ $n = 7$ ### Exam Strategy & Shortcut When you see $2^{n-1} + 2^{n+1}$, recognize mentally that this factors to $2^{n-1}(1 + 4) = 5 \times 2^{n-1}$. Instantly divide $320$ by $5$ to get $64$. Knowing your powers of $2$ is crucial for aptitude exams; recognizing $64$ as $2^6$ means $n-1 = 6$, giving $n = 7$ almost entirely mentally. ### Common Pitfall A major pitfall is improperly factoring the expression, often resulting in students forgetting the "1" when factoring out a term entirely (e.g., writing $2^{n-1}(2^2)$ instead of $2^{n-1}(1 + 2^2)$). Always double-check by mentally distributing the factored term back. ### Final Answer **Therefore, the correct answer is 7.**