Find $n$ if $7^{3n-1} - 343^{\,n-1} = 2352$.
Answer and solution
Correct answer: D
Solution
$343 = 7^3$, so $343^{\,n-1} = 7^{3n-3}$
$7^{3n-1} - 7^{3n-3} = 7^{3n-3}(7^2 - 1) = 48 \cdot 7^{3n-3} = 2352$
$7^{3n-3} = 49 = 7^2 \Rightarrow 3n - 3 = 2 \Rightarrow n = \dfrac{5}{3}$
$7^{3n-1} - 7^{3n-3} = 7^{3n-3}(7^2 - 1) = 48 \cdot 7^{3n-3} = 2352$
$7^{3n-3} = 49 = 7^2 \Rightarrow 3n - 3 = 2 \Rightarrow n = \dfrac{5}{3}$