Key takeaway: An arithmetic series adds numbers separated by a fixed step, and its total equals the count of terms multiplied by the midpoint of the two ends: S_n = n(a₁ + aₙ)/2.
Grade 11 – Algebra 2A-SSE.B.4Expressions and Equations
Definition of Arithmetic Series
An arithmetic series appears whenever you total a list of numbers that climb (or fall) by one fixed step. The list 14, 17, 20, 23 moves by 3 each time, so \(14+17+20+23\) qualifies.
Adding a long list term by term is tedious. Pairing rescues you: the two ends of the list total the same as the next two inward, and so on down the middle. Every pair matches, so the grand total is the number of pairs times one pair sum. Young Gauss is said to have used this trick.
Formula and notation
Let the opening value be \(a_1\), the step \(d\), and the count \(n\):
\[a_n=a_1+(n-1)d, \quad S_n=\frac{n\,(a_1+a_n)}{2}.\]
Equivalently, \(S_n=\frac{n}{2}\bigl(2a_1+(n-1)d\bigr)\), which skips the step of finding the last term.
Key facts
- Total = how many terms, times the midpoint of the smallest and largest term.
- A series with a constant difference of 0 is just n copies of one number.
- The last term is a₁ + (n − 1)d, one step fewer than n steps.
- Partial sums of an arithmetic series grow like a quadratic function of n.
Arithmetic Series example
A ballpark seating section is built with 22 rows. Row one seats 14 fans, and every later row seats 3 more fans than the one in front of it. What is the section’s capacity?
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Last row: 21 steps of 3 beyond the first, so \(a_{22}=14+21\cdot 3=77\).
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Plug in: \(S_{22}=\frac{22\,(14+77)}{2}\).
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Simplify: \(11\times 91=1001\).
The section holds 1,001 seats.
Common mistake
Watch out
Bao computes the final row as \(14+22\cdot 3=80\) and reports \(11\times 94=1034\) seats.
Why it is wrong
Row one is already the first term, so only 21 steps of size 3 are left: 77, not 80. Counting \(n\) steps instead of \(n-1\) overshoots the arithmetic series by one difference per pair, 33 seats in all.
Related terms and lessons
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