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Arithmetic Sequence Formula
Algebra • Grades 8, 9, 10, 11
Algebra
Arithmetic Sequence Formula
Find the n-th term (aₙ) and sum (Sₙ) of any linear arithmetic sequence.
The Formula
aₙ = a₁ + (n - 1)d
Used for: Finding the n-th term (aₙ) and sum of first n terms (Sₙ) in any linear arithmetic sequence
aₙn-th term in the sequence
a₁First term of the sequence
dCommon difference: (a₂ - a₁)
nTerm position index
SₙSum of first n terms: (n/2)(a₁ + aₙ)
⚡Live Interactive Quick Solver
Instant Calculation10-th Term (a_10):48
Sum of first 10 terms (S_10): 255• a_10 = 3 + (10 - 1) × 5 = 48
a_n = a_1 + (n - 1)d, \quad S_n = \frac{n}{2}(2a_1 + (n - 1)d)
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🎯 Key Concepts & Rules
Essential principles for calculating with this formula:
🪜
aₙ = a₁ + (n - 1)dCommon difference d is added each step.
➕
Sₙ = (n/2)(a₁ + aₙ)Gauss summation formula for first n terms.
📝 Worked Examples & Step-by-Step Solutions
Example 1: Sequence 3, 7, 11, 15... (Find 20th Term)
Problem:
First term a₁ = 3, common difference d = 4. Find a₂₀.1
Formula
a₂₀ = 3 + (20 - 1) × 42
Calculate
a₂₀ = 3 + 19 × 4 = 3 + 76 = 79Answer: a₂₀ = 79
💡 Pro Tips & Common Mistakes
Finding Common Difference
d = a₂ - a₁ = a₃ - a₂.
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