Arithmetic Sequences

An arithmetic sequence is a list of numbers that changes by the same fixed amount from one term to the next. Two short formulas handle almost every question about such a sequence: one finds any single term directly from its position, and the other finds the sum of many consecutive terms without adding them one at a time.

What you will learn

• Define an arithmetic sequence by its common difference
• Find a term using the position formula
• Sum a finite arithmetic sequence by pairing terms

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Recognizing an arithmetic sequence

A sequence is arithmetic when the difference between every pair of consecutive terms is the same fixed number, called the common difference. In $3,7,11,15,\ldots$, each term is 4 more than the one before it, so the common difference is $d=4$. Checking a sequence for this property means subtracting each term from the next and confirming every result is identical.

The common difference can be negative, which produces a decreasing sequence, such as $20,17,14,11,\ldots$ with $d=-3$. A sequence that changes by a different amount at each step, or by multiplying rather than adding, is not arithmetic.

Finding a term directly from its position

Rather than adding the common difference repeatedly to reach a far-off term, the position formula finds it directly: the $n$th term is $a_n=a_1+(n-1)d$, where $a_1$ is the first term. This works because reaching the $n$th term from the first term requires exactly $n-1$ steps of size $d$, not $n$ steps — a common off-by-one error to watch for.

The same formula runs in reverse to find a term's position when its value is known: setting $a_n$ equal to the given value and solving for $n$ answers questions like "which term of the sequence equals 50?"

Summing by pairing terms

The sum of a finite arithmetic sequence can be found without adding every term individually, using the same pairing trick famously attributed to a young Gauss: pairing the first term with the last, the second term with the second-to-last, and so on, always produces the same pair sum, $a_1+a_n$. Since there are $n$ terms total, there are $\frac{n}{2}$ such pairs, giving the sum formula $S_n=\frac{n(a_1+a_n)}{2}$.

Applying this formula requires knowing the number of terms $n$, which is not always given directly — it can be found from the position formula as $n=\frac{a_n-a_1}{d}+1$ whenever the first term, last term, and common difference are all known.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Using $n$ steps of the common difference instead of $n-1$ steps when finding a term from its position.
• Averaging the first and last terms but forgetting to multiply by the number of terms when finding a sum.
• Miscounting the number of terms in a sequence that does not start at 1, such as multiples of 5 starting at 5 rather than 0.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• An arithmetic sequence adds the same common difference $d$ to get from each term to the next.
• The $n$th term is $a_n=a_1+(n-1)d$, using $n-1$ steps from the first term.
• The sum of $n$ terms is $S_n=\frac{n(a_1+a_n)}{2}$, found by pairing the first and last terms.