Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 sequences and series? Then this Algebra 2 lesson on Sequences and Series: Patterns That Add Up explains each idea step by step, with worked examples.
Quick answer: An arithmetic sequence adds a constant d to each term, so a_n = a_1 + (n − 1)d, while a geometric sequence multiplies by a constant ratio r, so a_n = a_1 · r^(n−1). A series is a sum of terms. Finite sums are n(a_1 + a_n)/2 for arithmetic and a_1(1 − r^n)/(1 − r) for geometric series.
Standards: A-SSE.B.4, F-BF.A.2, F-IF.A.3 · Level: Grade 11 Algebra 2
What are arithmetic and geometric sequences?
Think of a sequence as a numbered list. Move from one term to the next by adding a fixed d (arithmetic) or by multiplying by a fixed r (geometric).
| Type | Explicit | Recursive |
|---|---|---|
| Arithmetic | an = a1 + (n − 1)d | an = an−1 + d |
| Geometric | an = a1 · rn−1 | an = r · an−1 |
The recursive rule needs a starting value, such as a1 = 9. Explicit rules jump straight to any term.
What does sigma notation mean?
The symbol ∑ means “add up.” The index below it gives the first value, the number above gives the last, and the expression is evaluated for each whole number between them.
How do you add the terms of a series?
- Arithmetic: Sn = n(a1 + an)2
- Geometric: Sn = a1 · 1 − rn1 − r (r ≠ 1)
- Identify the type: constant difference or constant ratio.
- Find n by solving the explicit rule if it is not given.
- Substitute into the matching sum formula.
Find the sum 4 + 6 + 9 + 13.5 + 20.25 + 30.375. The ratio is 6 ÷ 4 = 1.5, n = 6. S6 = 4 · 1 − 1.561 − 1.5 = 4 · −10.390625−0.5 = 83.125.
Using n instead of n − 1 in the exponent of the explicit geometric rule. The first term has exponent 0.
Where do series appear in money problems?
Saving the same amount each month earns interest on every deposit, so the balance is a geometric series. A loan payment is found by setting the sum of the discounted payments equal to the amount borrowed.
Before using any formula, write the first three terms. If you can see what happens each step, you will pick the right formula.
- Arithmetic: constant difference d; geometric: constant ratio r.
- Explicit rules give any term directly; recursive rules need the previous term and a starting value.
- Sigma notation ∑ adds an expression from a starting index to an ending index.
- Arithmetic sum: Sn = n(a1 + an)/2.
- Geometric sum: Sn = a1(1 − rn)/(1 − r), with r ≠ 1.
Sequences and Series: Patterns That Add Up: frequently asked questions
How do I tell whether a sequence is arithmetic or geometric?
Subtract consecutive terms. If the differences are all equal it is arithmetic. If not, divide consecutive terms. If the ratios are all equal it is geometric. Some sequences are neither.
What is the difference between a sequence and a series?
A sequence is a list of terms, such as 3, 7, 11. A series is the sum of those terms, such as 3 + 7 + 11 = 21. Sigma notation describes a series.
Why is the exponent n − 1 in the geometric rule?
The first term has not been multiplied by r yet, so it carries r0. The second term has been multiplied once, giving r1, and the nth term has been multiplied n − 1 times.
How do I find the number of terms n?
Set the explicit rule equal to the last term and solve for n. For a geometric sequence use logarithms if the unknown is in an exponent. The result must be a whole number.
How are series used for savings plans?
Each monthly deposit grows for a different number of months. The balances form a geometric series with ratio 1 + i, where i is the monthly rate, and the geometric sum formula gives the total.
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