Arithmetic Sequence And Series

A sequence is a list; a series is the sum of that list. For arithmetic patterns, that one distinction decides which formula you reach for — the term formula or the sum formula.

An arithmetic sequence changes by the same amount each step (the common difference $d$), and an arithmetic series is the sum of its terms. With first term $a_1$ and common difference $d$:

Sequence vs. series, side by side

                  Arithmetic sequence        Arithmetic series
Question asked    "What is a term?"          "What is the total?"
Object            ordered list itself        sum of the list's terms
Formula           a_n = a_1 + (n-1)d         S_n = (n/2)(a_1 + a_n)
Example: 2,5,8,11 the list 2, 5, 8, 11       2 + 5 + 8 + 11 = 26

A sequence is arithmetic only if the difference between consecutive terms stays constant. For instance, $4, 7, 10, 13, 16$ is arithmetic because each term increases by $3$, so $d = 3$. By contrast $5, 9, 14, 20$ is not, because the differences are $4$, $5$, and $6$; since the difference changes, the arithmetic formulas do not apply.

When to use which

Use the term formula $a_n = a_1 + (n-1)d$ when a question asks for a particular term, such as the $20$th value. Use the sum formula $S_n = \frac{n}{2}(a_1 + a_n)$ when a question asks for the total of the first $n$ terms; if you do not yet know the last term $a_n$, find it first with the term formula. Both require a constant common difference.

Worked example: the $20$th term and the first $20$-term sum

Consider the sequence

so $a_1 = 5$ and $d = 3$.

Find the $20$th term with the term formula:

Find the sum of the first $20$ terms with the sum formula, using $a_{20} = 62$:

So the $20$th term is $62$ and the sum of the first $20$ terms is $670$. The sum formula works because the first and last terms have the same average as the second and second-to-last, and so on inward; the total is therefore the number of terms times the average of the first and last term.

Common confusions and how to keep them straight

• Mixing up $n$ and $d$. $n$ counts the position or number of terms; $d$ is the constant difference. Different jobs.
• Forgetting the $(n-1)$. The term formula is $a_n = a_1 + (n-1)d$, not $a_1 + nd$, because there are only $n-1$ jumps from the first term to the $n$th.
• Using the sum formula on a non-arithmetic list. If the differences are not constant, the arithmetic series formula does not apply — check the pattern first.
• Losing the sign of the difference. A decreasing sequence has negative $d$: in $12, 9, 6, 3$, the common difference is $-3$, not $3$.

When arithmetic sequences and series are used

They appear whenever a quantity changes by a constant amount each step: saving the same amount monthly, rows of seats that grow by a fixed number, algebra problems built on linear growth. They fit additive change. If each step multiplies by the same factor instead of adding the same amount, you have a geometric sequence instead.

To compare the two ideas directly, take $18, 15, 12, 9, \ldots$ and find the common difference, the $12$th term, and the sum of the first $12$ terms. Then solve the same kind of problem for a geometric sequence and notice what changes when the pattern is constant multiplication rather than constant addition.