The given equation is:

$(7)^{\frac{n-1}{3}} + 6 = 7$

We need to solve for $n$.

Step-by-Step Solution:

1. Isolate the exponential term: $(7)^{\frac{n-1}{3}} + 6 = 7$ Subtract 6 from both sides: $(7)^{\frac{n-1}{3}} = 1$

2. Express the equation in logarithmic or exponential form: We know that $7^0 = 1$, so: $\frac{n-1}{3} = 0$

3. Solve for $n$: Multiply both sides of the equation by 3: $n - 1 = 0$ Add 1 to both sides: $n = 1$

Thus, the solution for $n$ is $1$.

However, none of the options in the answer choices match this solution directly. Let me know if you'd like further clarification or analysis of the options!

Would you like to dive deeper into any step?

Here are some related questions to explore:

1. What are the properties of exponents and logarithms that are useful in solving equations like this?
2. Can we generalize this type of problem for different bases other than 7?
3. How can we solve exponential equations when the base is different on both sides?
4. How can logarithms be used to solve more complicated exponential equations?
5. What are common mistakes when solving exponential equations like this one?

Tip: Always check the constraints of the base and exponent to avoid overlooking solutions in complex cases.