Mathematics

The key insight is that the sum formula $S_n = \frac{n}{2}(2a + (n-1)d)$ must simplify to a linear function in $n$ for proportionality to hold. When you expand this, you get a quadratic term in $n$ unless the coefficient of $n^2$ vanishes—which forces the common difference $d$ to equal zero. This means the progression must be constant, where every term equals the first term $a$.

The key insight is that the sum formula $S_n = \frac{n}{2}(2a + (n-1)d)$ must simplify to a linear function in $n$ for proportionality to hold. When you expand this, you get a quadratic term in $n$ unless the coefficient of $n^2$ vanishes—which forces the common difference $d$ to equal zero. This means the progression must be constant, where every term equals the first term $a$.