Arithmetic Sequence Calculator
Calculate terms, common difference, sum, and properties of arithmetic sequences and series with step-by-step solutions.
Find the n-th term using: aₙ = a₁ + (n − 1)d
Find the sum of first n terms: Sₙ = n/2 × (2a₁ + (n − 1)d)
Find the common difference from two terms: d = (aₘ − aₖ) / (m − k)
Find how many terms: n = (aₙ − a₁) / d + 1
Result
Sequence Properties
Sequence Terms
Step-by-Step Solution
Understanding Arithmetic Sequences
An arithmetic sequence (also called an arithmetic progression or AP) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by d.
General Term Formula
The n-th term of an arithmetic sequence is:
Where a₁ is the first term, d is the common difference, and n is the position of the term.
Sum of an Arithmetic Series
The sum of the first n terms (called an arithmetic series) can be computed using two equivalent formulas:
Sₙ = n/2 × (2a₁ + (n − 1)d)
Sₙ = n/2 × (a₁ + aₙ)
The second form is especially intuitive: the sum equals the number of terms multiplied by the average of the first and last terms.
Common Difference
The common difference can be found from any two terms:
If d > 0 the sequence is increasing, if d < 0 it is decreasing, and if d = 0 all terms are equal (constant sequence).
Key Properties
Arithmetic Mean
Every term in an AP (except the first and last) is the arithmetic mean of its two neighbors: aₙ = (aₙ₋₁ + aₙ₊₁) / 2. This property can be used to verify that a sequence is arithmetic.
Constant Second Difference
In an AP the first differences are all equal to d. If a sequence has constant first differences, it is arithmetic. Quadratic sequences, in contrast, have constant second differences.
Linear Representation
Plotting an AP against position gives a straight line with slope d and y-intercept (a₁ − d). The general term formula aₙ = dn + (a₁ − d) is a linear function of n.
Gauss's Sum Trick
The sum formula Sₙ = n(a₁ + aₙ)/2 is attributed to young Gauss, who famously summed 1 + 2 + ... + 100 = 5050 by pairing terms from opposite ends. Each pair sums to 101, and there are 50 pairs.
Common Examples
Example 1: Finding the 20th Term
Sequence: 3, 7, 11, 15, ... (a₁ = 3, d = 4)
Formula: a₂₀ = 3 + (20 − 1) × 4 = 3 + 76 = 79
Example 2: Sum of First 50 Natural Numbers
Sequence: 1, 2, 3, ..., 50 (a₁ = 1, d = 1, n = 50)
Formula: S₅₀ = 50/2 × (1 + 50) = 25 × 51 = 1275
Example 3: Finding the Common Difference
Given: a₃ = 10 and a₇ = 26
Formula: d = (26 − 10) / (7 − 3) = 16 / 4 = 4
First term: a₁ = a₃ − 2d = 10 − 8 = 2
Example 4: Sum of Even Numbers from 2 to 100
Sequence: 2, 4, 6, ..., 100 (a₁ = 2, d = 2, aₙ = 100)
Number of terms: n = (100 − 2) / 2 + 1 = 50
Sum: S₅₀ = 50/2 × (2 + 100) = 25 × 102 = 2550
Real-World Applications
- Finance: Fixed installment payments, straight-line depreciation, and equal periodic savings deposits
- Physics: Uniformly accelerated motion (distance traveled in successive equal time intervals forms an AP)
- Architecture: Evenly spaced columns, stair steps with uniform rise, and seating row arrangements
- Calendar math: Dates falling on the same day of the week (every 7 days) form an AP
- Computer science: Loop counters, memory address calculations, and array indexing
- Music: Equally tempered pitch intervals and rhythmic patterns
References
The formulas and concepts used in this calculator are based on well-established mathematical principles:
Related Calculators
This calculator computes arithmetic sequence properties using standard formulas. Results are for educational and informational purposes. While we strive for accuracy, please verify important calculations independently.