Algebra 2 Unit 8 study games — Sequences and Series.
This unit covers arithmetic sequences, geometric sequences and sigma notation — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers arithmetic sequences, geometric sequences and sigma notation — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Arithmetic Sequences
An arithmetic sequence has a constant difference (common difference, d) between consecutive terms. You must be able to find any term using the explicit formula and identify whether a sequence is arithmetic. Exams test finding missing terms, writing formulas, and solving for n given a term value.
Key Points
- Explicit formula: a_n = a_1 + (n - 1)d
- Common difference d = a_n - a_(n-1) (subtract any term from the next)
- If given two terms, set up a system or use d = (a_m - a_n) / (m - n)
- The graph of an arithmetic sequence is linear — points lie on a straight line
The 3rd term of an arithmetic sequence is 11 and the 7th term is 27. Find a_1 and write the explicit formula.
First find d: d = (27 - 11) / (7 - 3) = 16 / 4 = 4. Then use a_3 = a_1 + (3-1)d to get 11 = a_1 + 8, so a_1 = 3. The explicit formula is a_n = 3 + (n - 1)(4), which simplifies to a_n = 4n - 1.
2 Geometric Sequences
A geometric sequence has a constant ratio (common ratio, r) between consecutive terms found by dividing any term by the one before it. You must know both the explicit formula and how to identify geometric sequences. Exams also test recognizing when r is a fraction (decay) versus r > 1 (growth).
Key Points
- Explicit formula: a_n = a_1 · r^(n - 1)
- Common ratio r = a_n / a_(n-1) (divide any term by the previous term)
- If r is between -1 and 1 (exclusive), terms approach zero; if |r| > 1, terms grow without bound
- Geometric means: to insert k means between two terms, find r = (a_last / a_first)^(1/(k+1))
A geometric sequence has a_1 = 5 and r = 3. Which term equals 1215?
Set the explicit formula equal to 1215: 5 · 3^(n-1) = 1215. Divide both sides by 5 to get 3^(n-1) = 243. Recognize that 3^5 = 243, so n - 1 = 5, meaning n = 6. The 6th term is 1215.
3 Sigma Notation
Sigma notation (Σ) is a compact way to write a sum; you must be able to expand it into individual terms and evaluate it. Exams test reading the index, limits, and expression correctly, as well as applying arithmetic or geometric series sum formulas. Know both sum formulas and when each applies.
Key Points
- Σ from k=1 to n of a_k means add a_1 + a_2 + ... + a_n; the variable under Σ is the index
- Arithmetic series sum: S_n = n/2 · (a_1 + a_n) or S_n = n/2 · [2a_1 + (n-1)d]
- Geometric series sum: S_n = a_1 · (1 - r^n) / (1 - r), where r ≠ 1
- Always identify whether the series is arithmetic or geometric before choosing a formula
Evaluate: Σ (k=1 to 6) of 3 · 2^(k-1).
This is a geometric series with a_1 = 3 · 2^0 = 3 and r = 2, summing 6 terms. Apply the geometric sum formula: S_6 = 3 · (1 - 2^6) / (1 - 2) = 3 · (1 - 64) / (-1) = 3 · 63 = 189.
Questions, answered.
What is Sequences and Series?
Sequences and Series is Unit 8 of Algebra 2, covering arithmetic sequences, geometric sequences and sigma notation.
How to study for Algebra 2 Unit 8?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 27+ review questions across 5 different game modes.