We are tasked with proving the formula for the sum of the arithmetic sequence $S(n) = 1 + 4 + 7 + 10 + \dots + (3n + 1)$ using mathematical induction. Specifically, we aim to show that:

$S(n) = \frac{(3n + 2)(n + 1)}{2}$

Step 1: Base Case (n = 1)

For $n = 1$, the left-hand side (LHS) of the equation is just the first term of the sequence:

$S(1) = 1$

Now check the right-hand side (RHS) of the equation for $n = 1$:

$\frac{(3(1) + 2)(1 + 1)}{2} = \frac{(3 + 2)(2)}{2} = \frac{5 \times 2}{2} = 5$

Clearly, the base case does not seem to hold since $1 \neq 5$. However, let's reconsider the sequence starting from $n = 0$. If we start with $n = 0$, the sequence becomes clearer, and the base case will likely be different.

Let me check that now. Should I expand starting from zero?