Question:

Let the given expression be $P(n) = 10^n + 3(4^n + 2) + 5$. First, simplify the expression: $$P(n) = 10^n + 3 \cdot 4^n + 6 + 5$$ $$P(n) = 10^n + 3 \cdot 4^n + 11$$ To determine divisibility by 7, we analyze the expression modulo 7. We note the following congruences: $$10 \equiv 3 \pmod{7}$$ $$4 \equiv 4 \pmod{7}$$ $$11 \equiv 4 \pmod{7}$$ Substitute these congruences into the simplified expression for $P(n)$: $$P(n) \equiv 3^n + 3 \cdot 4^n + 4 \pmod{7}$$ Now, evaluate the expression for $n=1$: $$P(1) \equiv 3^1 + 3 \cdot 4^1 + 4 \pmod{7}$$ $$P(1) \equiv 3 + 12 + 4 \pmod{7}$$ Since $12 \equiv 5 \pmod{7}$, we have: $$P(1) \equiv 3 + 5 + 4 \pmod{7}$$ $$P(1) \equiv 12 \pmod{7}$$ $$P(1) \equiv 5 \pmod{7}$$ Since $P(1) \equiv 5 \pmod{7}$, the expression $P(n)$ is not divisible by 7 for $n=1$. Therefore, the expression $10^n + 3(4^n + 2) + 5$ is not divisible by 7 for all $n \in \mathbb{N}$.