🎓 Mathematics • Sequences & Series
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Sequence & Series Calculator
Solves arithmetic sequences (an = a1 + (n-1)d) and geometric sequences (an = a1·r^(n-1)) with partial sums.
Arithmetic Progression Parameters
AP Quick Presets:
Progressive Solution (n = 10)
10-th Term (a_10)
48
aₙ = a₁ + (n - 1)dSum of First 10 Terms (S_10)
255
Sₙ = (n/2)(a₁ + aₙ)Sequence Terms:
a_1:3
a_2:8
a_3:13
a_4:18
a_5:23
a_6:28
a_7:33
a_8:38
a_9:43
a_10:48
Mathematical Derivation & Formulas:
1. N-th Term Formula: aₙ = a₁ + (n - 1)d
= 3 + (10 - 1) × 5 = 3 + 45 = 48
2. Sum Formula: Sₙ = (n / 2) × (a₁ + aₙ)
= (10 / 2) × (3 + 48) = 5 × 51 = 255
What are Sequences and Series?
A sequence is an ordered list of numbers following a rule. A series is the sum of the terms of a sequence.
Formula & Step-by-Step Calculation
Arith: aₙ = a₁ + (n-1)d, Geo: aₙ = a₁ · rⁿ⁻¹
Formulas for arithmetic common difference d and geometric common ratio r.
Worked Step-by-Step Examples
Example 1
10th term of 3, 8, 13, 18...
Solution: a₁₀ = 48
• a₁ = 3, d = 5. a₁₀ = 3 + (9 × 5) = 3 + 45 = 48.
Common Real-World & Academic Use Cases
- ✓ Compound savings annuity accumulations
- ✓ Radioactive half-life series
- ✓ Pattern recognition in mathematics
How to Use the Sequence & Series Calculator
1
Select Type
Choose Arithmetic or Geometric.
2
Input a₁, d/r, n
Enter initial parameters.
3
Inspect n-th Term & Sum
View term value and summation.
Frequently Asked Questions
Q: When does an infinite geometric series converge?
Only when the common ratio satisfies |r| < 1, converging to S = a / (1 - r).