Arithmetic sequence (n-th term)

What the symbols mean

$a_n$

the n-th term: the one you want

$a_1$

the first term

$d$

the common difference: what you add each step (negative if the sequence goes down)

$n$

the position of the term (1 for the first, 2 for the second, …)

When to use it

Use it whenever consecutive terms differ by a constant: seat rows that grow by 2 each row, a salary that rises by a fixed amount a year, the house numbers along one side of a street. Solve for n to check whether a number belongs to the sequence and which term it is, or for d when you know two terms.

Why it works

Getting from the first term to the n-th takes $n-1$ steps, and each step adds d. That is why the multiplier is $n-1$, not n: the first term has had no steps added yet.

Common mistakes

• Multiplying d by n instead of $n-1$: the 10th term is $3+9\times 4=39$, not $3+10\times 4$.
• Getting the sign of d wrong for a decreasing sequence: 20, 17, 14, … has d = −3.
• Accepting a fractional n. If solving for n gives 9.25, the number is not in the sequence at all.