The Off-by-One Trap — Sequences & Series (Mathematics)

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Watch Out

Off-by-One: n vs (n−1)

The most common sequence error. The nth-term formula uses $(n-1)$, not $n$. Between the 1st and 5th term, you take only 4 steps — not 5. Off-by-one is always a trap option.

n terms ⟹ (n - 1) steps between them

Counting fence posts vs counting gaps

1 to 5

= 5 terms

But only 4 differences between them

$a_{n} = a_{1} + (n - 1) d$

Not $n \cdot d$ !

Skip the $(n - 1)$ → off by one $d$

Sanity check

Plug $n = 1$

Should give back $a_{1}$ . If not, your formula is wrong.