ACT® · Math: Functions — Sequences and Series
Summing an Arithmetic Series
To sum an arithmetic series, find the average of the first and last term, then multiply by the number of terms. That single move replaces adding every term by hand.
In an arithmetic sequence each term increases by the same common difference, so the terms are evenly spaced. That even spacing means pairs from opposite ends always add to the same value: first plus last, second plus second-to-last, and so on all give that one fixed sum. Once every pair totals the same amount, the whole series collapses into a multiplication instead of a long addition.
The formula says the sum equals the average of the first and last term, times the number of terms: S = n(a1 + an)/2. The average of the two ends tells you what a 'typical' term is worth, and multiplying by how many terms there are scales that typical value up to the full total. You never need to know every term in between, only where the sequence starts, where it ends, and how many terms it has.
This sits right next to the explicit formula for the nth term, an = a1 + (n-1)d, which you often need first to find the last term before you can sum anything. Together the two formulas let you jump from a recursive rule straight to a total, without listing terms. The same pairing logic will not work for geometric series, where the terms are not evenly spaced, so geometric sums need a different formula built around the common ratio instead.
Key terms
- Arithmetic sequence
- A list of numbers where each term is the previous term plus a fixed common difference.
- Common difference
- The constant amount added to each term to get the next term, often called d.
- Explicit formula
- A rule giving any term directly from its position, an = a1 + (n-1)d.
- Series
- The sum of the terms of a sequence, rather than the list of terms itself.
Watch it explained
Check yourself
End of topic.