To solve this equation, we can first simplify the left-hand side:
2^(3n+1) / (2^n - 1) = 3n + 16
We can then multiply both sides by 2^n - 1 to eliminate the denominator:
2^(3n+1) = (3n + 16)(2^n - 1)
Expanding the right-hand side:
2^(3n+1) = 6n*2^n + 16*2^n - 3n - 16
Simplifying:
2^(3n+1) = (6n - 3n)*2^n + (16 - 16)
2^(3n+1) = 3n*2^n
Dividing both sides by 2^n:
2^(n+1) = 3n
Taking the logarithm of both sides (with any base), we get:
n log(2) + log(2) = log(3n)
n = (log(3n) - log(2)) / log(2)
We can use a calculator to evaluate this expression, or simplify it further using logarithm rules:
n = log2(3n/2)
We can now use iterative methods or a graphing calculator to find a numerical solution for n. The solution is approximately 7.524.