Traps in Sequence & Series

Using $a_n = a + nd$ instead of $a + (n-1)d$. The 1st term is $a$, not $a + d$.

Why: The index starts at 1, not 0, so the first term has zero common differences added.

The GP sum formula $S_n = \frac{a(r^n - 1)}{r - 1}$ is undefined when $r = 1$.

Why: The formula has $(r - 1)$ in the denominator, which is zero when $r = 1$.

$S_\infty = \frac{a}{1 - r}$ only works when $|r| < 1$. Series diverges otherwise.

Why: Students memorize the formula without the convergence condition.

Students sometimes use the $S_n$ formula when asked for $a_n$, or vice versa.

Why: Both involve the same variables $a$, $d$, and $n$, so the formulas look similar.

When computing $T_n = S_n - S_{n-1}$, students mess up the algebra while expanding.

Why: Expanding $(n-1)^2$ requires careful distribution of the negative sign across all terms.