Arithmetic Sequence Calculator
Arithmetic Sequence Solver
Calculate nth term, sequence sum, common difference, and generate arithmetic progression with step-by-step solutions.
Sequence Result
29
Sequence Terms:
Formula Used:
aₙ = a₁ + (n-1)d
Calculation:
a₁₀ = 2 + (10-1)×3 = 29
Properties:
Increasing sequence, d > 0
Step-by-Step Solution:
Arithmetic sequence: Each term differs by a constant amount called common difference.
What is an Arithmetic Sequence?
An arithmetic sequence (also called arithmetic progression) is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference (d). The sequence follows a linear pattern, making it one of the simplest and most fundamental mathematical sequences.
Key Formulas
Nth Term Formula
Find any term in the sequence
Sum of First n Terms
Or: Sₙ = n/2 × (a₁ + aₙ)
Common Difference
Constant difference between terms
Number of Terms
Find total terms in sequence
Types of Arithmetic Sequences
1. Increasing Sequences (d > 0)
When the common difference is positive, each term is larger than the previous:
Example: 2, 5, 8, 11, 14... (d = 3)
Pattern: Each term increases by 3
2. Decreasing Sequences (d < 0)
When the common difference is negative, each term is smaller than the previous:
Example: 20, 17, 14, 11, 8... (d = -3)
Pattern: Each term decreases by 3
3. Constant Sequences (d = 0)
When the common difference is zero, all terms are equal:
Example: 5, 5, 5, 5, 5... (d = 0)
Pattern: All terms are identical
Real-World Applications
Finance & Business
- Loan payments: Equal monthly installments form arithmetic sequence
- Depreciation: Straight-line depreciation follows arithmetic pattern
- Salary increments: Fixed annual salary increases
- Inventory management: Regular restocking schedules
Science & Engineering
- Physics: Objects moving with constant acceleration
- Computer science: Array indexing and memory allocation
- Architecture: Regular spacing of columns or windows
- Music theory: Equal temperament tuning intervals
Everyday Life
- Saving money: Fixed monthly savings contributions
- Exercise plans: Adding fixed increments to workout routines
- Construction: Regular spacing of fence posts or tiles
- Time management: Scheduling events at regular intervals
Common Examples
| Sequence | First Term (a₁) | Common Diff (d) | 10th Term (a₁₀) | Sum of 10 Terms (S₁₀) |
|---|---|---|---|---|
| Natural Numbers | 1 | 1 | 10 | 55 |
| Even Numbers | 2 | 2 | 20 | 110 |
| Odd Numbers | 1 | 2 | 19 | 100 |
| Multiples of 5 | 5 | 5 | 50 | 275 |
Step-by-Step Examples
Example 1: Find the 15th term of sequence 3, 7, 11, 15...
- Identify first term: a₁ = 3
- Find common difference: d = 7 - 3 = 4
- Use nth term formula: aₙ = a₁ + (n-1)d
- Plug values: a₁₅ = 3 + (15-1)×4
- Calculate: a₁₅ = 3 + 14×4 = 3 + 56 = 59
- Answer: The 15th term is 59
Example 2: Find sum of first 20 terms where a₁ = 5, d = 2
- Identify values: a₁ = 5, d = 2, n = 20
- Use sum formula: Sₙ = n/2 × [2a₁ + (n-1)d]
- Plug values: S₂₀ = 20/2 × [2×5 + (20-1)×2]
- Calculate inside: 2×5 + 19×2 = 10 + 38 = 48
- Complete: S₂₀ = 10 × 48 = 480
- Answer: Sum of first 20 terms is 480
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Frequently Asked Questions (FAQs)
Q: What's the difference between arithmetic and geometric sequence?
A: Arithmetic sequence adds a constant difference (d) between terms, while geometric sequence multiplies by a constant ratio (r). Example: Arithmetic: 2,5,8,11... (add 3); Geometric: 2,6,18,54... (multiply by 3).
Q: How do I find the common difference if I know two terms?
A: Use formula: d = (aₙ - aₘ) / (n - m), where aₙ and aₘ are terms at positions n and m. Example: If 3rd term is 10 and 7th term is 22, then d = (22-10)/(7-3) = 12/4 = 3.
Q: Can an arithmetic sequence have fractional common difference?
A: Yes! Common difference can be any real number: integer, fraction, decimal, positive, or negative. Example: ½, 1, 1½, 2, 2½... has d = 0.5.
Q: What is the sum of first n natural numbers?
A: Natural numbers 1,2,3,4... form arithmetic sequence with a₁=1, d=1. Sum formula: Sₙ = n(n+1)/2. For n=100, sum = 100×101/2 = 5050.
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