Grades Algebra II & Trigonometry Sequences, Series & MatricesArithmetic & Geometric Sequences

Arithmetic & Geometric Sequences

🔢 Sequences, Series & Matrices

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Arithmetic & Geometric Sequences

Find terms and sums of arithmetic and geometric sequences.

A sequence is an ordered list of numbers. An arithmetic sequence has a constant difference d between consecutive terms; a geometric sequence has a constant ratio r.

Arithmetic Sequence

aₙ = a₁ + (n−1)dCommon difference dSₙ = n/2 · (a₁ + aₙ)Example: 3, 7, 11, 15 (d = 4)

Geometric Sequence

aₙ = a₁ · rⁿ⁻¹Common ratio rSₙ = a₁(1 − rⁿ)/(1 − r)Example: 2, 6, 18, 54 (r = 3)

S_n^{\text{arith}} = \frac{n}{2}(a_1 + a_n) \qquad S_n^{\text{geo}} = \frac{a_1(1-r^n)}{1-r} \quad (r \ne 1)

∞

Infinite geometric series: if |r| < 1, the series converges to S = a₁/(1−r). If |r| ≥ 1, it diverges.

✏️Sum of First 10 Terms

Arithmetic: 5, 9, 13, … (d = 4). a₁₀ = 5 + 9(4) = 41. S₁₀ = 10/2 · (5+41) = 5 · 46 = 230. Geometric: 3, 6, 12, … (r = 2). S₁₀ = 3(1−2¹⁰)/(1−2) = 3(−1023)/(−1) = 3069.

✏️ Try It!

What is the sum of the infinite geometric series 8 + 4 + 2 + 1 + …?

Practice with CalcVerse

→Limit Simulator ∑Sequence & Series

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