Free online arithmetic sequence calculator to find terms, common difference, and sum of arithmetic progressions. Perfect for students learning sequences and series mathematics.
The starting value of your sequence
The constant amount added to each term
How many terms to calculate (max: 100)
Formula for nth term: aₙ = a₁ + (n - 1) × d
Formula for sum: Sₙ = n/2 × (2a₁ + (n - 1) × d)
Where: a₁ = first term, d = common difference, n = number of terms
An arithmetic sequence (or arithmetic progression) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference and is denoted by 'd'.
Nth Term Formula:
aₙ = a₁ + (n - 1) × d
Where aₙ is the nth term, a₁ is the first term, n is the term number
Sum of First n Terms:
Sₙ = n/2 × (2a₁ + (n - 1) × d)
Or: Sₙ = n/2 × (a₁ + aₙ)
The starting value of the sequence. This is the foundation from which all other terms are calculated by adding multiples of the common difference.
The constant amount added to each term to get the next term. Can be positive (increasing sequence), negative (decreasing), or zero (constant sequence).
The position of the term in the sequence you want to find. The first term is n=1, second term is n=2, and so on.
Find the 10th term of the sequence: 2, 5, 8, 11, 14...
First term (a₁): 2
Common difference (d): 5 - 2 = 3
Term to find (n): 10
a₁₀ = 2 + (10 - 1) × 3 = 2 + 27 = 29
Find the sum of first 8 terms: 20, 17, 14, 11, 8...
First term (a₁): 20
Common difference (d): 17 - 20 = -3
Number of terms (n): 8
S₈ = 8/2 × (2 × 20 + 7 × (-3)) = 4 × (40 - 21) = 4 × 19 = 76
A company saves $500 in the first month and increases savings by $100 each month. How much will they save in the 12th month?
First term (a₁): $500
Common difference (d): $100
Month to find (n): 12
a₁₂ = 500 + (12 - 1) × 100 = 500 + 1100 = $1600
Arithmetic sequences add a constant difference to get the next term, while geometric sequences multiply by a constant ratio. In arithmetic: 2, 5, 8, 11... (add 3 each time). In geometric: 2, 6, 18, 54... (multiply by 3 each time).
Yes! If the common difference is zero, all terms are equal (constant sequence). If negative, the sequence decreases. Both are valid arithmetic sequences that our calculator can handle.
Subtract any term from the next term: d = a₂ - a₁ = a₃ - a₂ = aₙ - aₙ₋₁. For example, in sequence 7, 12, 17, 22..., the common difference is 12 - 7 = 5.
The sum formula Sₙ = n/2 × (a₁ + aₙ) finds the total of all terms from first to nth. Use it when you need the total value, like total savings over months, total distance traveled, or total items produced.
Yes, our arithmetic sequence calculator supports large numbers, decimals, and negative values. It provides accurate results for both educational and real-world applications.
Our calculator provides highly accurate results using JavaScript's built-in mathematical operations. It's suitable for educational purposes, homework help, and practical applications.
Linear Growth:
Each term increases by a constant amount
Constant Difference:
aₙ₊₁ - aₙ = d (for all n)
Linear Formula:
aₙ = dn + (a₁ - d) is a linear function of n
Symmetric Sum:
First + last = second + second-last = ... = constant
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