The given problem asks us to prove by mathematical induction that the expression $3^n + 7^n - 2$ is divisible by 8 for $n \geq 1$. Let's proceed step by step.

Step 1: Base Case (n = 1)

We need to check if the statement is true for $n = 1$.

$3^1 + 7^1 - 2 = 3 + 7 - 2 = 8$

Since 8 is divisible by 8, the base case holds true.

Step 2: Inductive Hypothesis

Assume the statement is true for some $n = k$, where $k \geq 1$. That is:

$3^k + 7^k - 2 = 8m, \quad \text{for some integer } m.$

This means that $3^k + 7^k - 2$ is divisible by 8.

Step 3: Inductive Step (n = k + 1)

We need to prove that the statement holds for $n = k + 1$. That is, we need to show that:

$3^{k+1} + 7^{k+1} - 2$

is divisible by 8. Let's express $3^{k+1}$ and $7^{k+1}$ in terms of $3^k$ and $7^k$:

$3^{k+1} = 3 \cdot 3^k, \quad 7^{k+1} = 7 \cdot 7^k$

Now consider the expression:

$3^{k+1} + 7^{k+1} - 2 = 3 \cdot 3^k + 7 \cdot 7^k - 2$

Using the inductive hypothesis, we know that:

$3^k + 7^k - 2 = 8m \quad \text{(for some integer } m\text{)}$

To proceed further, let's look at the differences between $3^{k+1} + 7^{k+1} - 2$ and $3^k + 7^k - 2$:

$3^{k+1} + 7^{k+1} - 2 - (3^k + 7^k - 2)$

This