Moderate -0.3 This is a straightforward divisibility proof by induction with standard structure: verify base case n=1, assume for n=k, prove for n=k+1. The inductive step requires factoring out 7^k + 3^(k-1) and showing the remainder is divisible by 4 (using 7≡-1 mod 4 and 3≡-1 mod 4). While it requires careful algebraic manipulation, it follows a well-practiced template with no novel insights needed, making it slightly easier than average.
9. Prove by induction that, for all positive integers $n , 7 ^ { n } + 3 ^ { n - 1 }$ is a multiple of 4 .\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2024 Q9 [5]}}