1.5 Algebra Number Theory

Use mathematical induction to prove the formula for the sum of the first \( n \) terms of an arithmetic series, which is given by
$$ S_n = \sum_{k=1}^n (a + (k-1)d) = \frac{n}{2}(2a + (n-1)d) $$
for all positive integers \( n \ge 1 \), where \( a \) is the first term and \( d \) is the common difference.

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