Son güncelleme: 19 Ağustos 2026
Sequence Hesaplayıcı
Oluşturanlar
Dharmendra Singhİnceleyenler
Oluşturanlar
Dharmendra Singhİnceleyenler
Quick Answer
The sequence calculator solves two common cases: arithmetic and geometric sequences. It uses aₙ = a₁ + (n−1)d and Sₙ = n/2[2a₁ + (n−1)d] for arithmetic patterns, and aₙ = a₁r^(n−1) with Sₙ = a₁(1−rⁿ)/(1−r) for geometric patterns, while also previewing the first five terms for a quick setup check.
Use this sequence calculator to choose arithmetic or geometric mode, then find the n-th term and the sum of the first n terms from the same inputs.
Önemli Noktalar
- This calculator is scoped to arithmetic and geometric sequences only.
- Arithmetic sequences use a fixed difference, while geometric sequences use a fixed ratio.
- The calculator returns both the n-th term and the finite sum through n.
- A preview of the first five terms helps catch setup mistakes early.
- The special geometric case r = 1 is handled with a simpler sum formula.
Oluşturanlar
Dharmendra Singhİnceleyenler
Oluşturanlar
Dharmendra Singhİnceleyenler
Formül
Arithmetic: aₙ = a₁ + (n−1)d, Sₙ = n/2[2a₁ + (n−1)d]; Geometric: aₙ = a₁r^(n−1), Sₙ = a₁(1−rⁿ)/(1−r)
Burada:
- a₁=First term
- d=Common difference for arithmetic sequences
- r=Common ratio for geometric sequences
- n=Requested term number
Çözülmüş örnekler
Arithmetic sequence with first term 3 and difference 2
Find the 5th term and the sum of the first 5 terms.
- 1Use aₙ = a₁ + (n−1)d = 3 + 4 × 2 = 11.
- 2Use Sₙ = n/2[2a₁ + (n−1)d] = 5/2[6 + 8] = 35.
- 3The first five terms are 3, 5, 7, 9, 11.
Geometric sequence with first term 2 and ratio 3
Find the 4th term and the sum of the first 4 terms.
- 1Use aₙ = a₁r^(n−1) = 2 × 3³ = 54.
- 2Use Sₙ = a₁(1−rⁿ)/(1−r) = 2(1−81)/(1−3) = 80.
- 3The first four terms are 2, 6, 18, 54.
Arithmetic sequence with a negative difference
A decreasing sequence still uses the same nth-term and sum formulas.
- 1Use a₆ = 10 + 5 × (−1.5) = 2.5.
- 2Use S₆ = 6/2[20 − 7.5] = 37.5.
- 3The first six terms are 10, 8.5, 7, 5.5, 4, 2.5.
Giriş
The Sequence Calculator covers the two most common classroom and pre-calculus cases in one place: arithmetic sequences and geometric sequences. You choose the type, enter the first term, the common difference or ratio, and the term number, and the page returns both the n-th term and the sum of the first n terms. By keeping the scope tight instead of trying to include every famous sequence ever invented, the calculator stays clear, fast, and easy to verify by hand.
What this sequence calculator covers
This calculator is intentionally scoped to arithmetic and geometric sequences. That means it handles patterns built from a fixed difference or a fixed ratio, which are the two formulas students and professionals use most often when forecasting repeated change or checking series totals.
Arithmetic sequences add the same amount each step.
Geometric sequences multiply by the same ratio each step.
The calculator returns both a term and a partial sum.
A short preview of terms helps check the pattern quickly.
Arithmetic sequence formulas explained
Arithmetic sequences grow or shrink by a fixed difference d. The explicit formula aₙ = a₁ + (n−1)d tells you any term directly, while Sₙ = n/2[2a₁ + (n−1)d] adds the first n terms without listing every one. These formulas are linear because the step change stays constant.
Use d when each term is found by adding or subtracting.
Negative d gives a decreasing arithmetic sequence.
The n-th term depends linearly on n.
The partial sum grows quadratically when d is nonzero.
Geometric sequence formulas explained
Geometric sequences use a common ratio r, so each term is the previous term multiplied by r. The term formula aₙ = a₁r^(n−1) captures that repeated multiplication. The finite-sum formula Sₙ = a₁(1−rⁿ)/(1−r) works whenever r is not 1, and the calculator uses the simpler Sₙ = na₁ special case when r = 1.
Use r when each step multiplies rather than adds.
If r = 1, every term stays the same.
A negative ratio makes signs alternate.
Geometric growth is exponential, not linear.
Worked examples you can verify by hand
The arithmetic example 3, 5, 7, 9, 11 leads to a fifth term of 11 and a first-five sum of 35. The geometric example 2, 6, 18, 54 leads to a fourth term of 54 and a first-four sum of 80. Those are good benchmark cases because the arithmetic is clean enough to check without software.
List a few terms first if you want a quick confidence check.
Use the closed formula after the pattern is clear.
The preview should match the sequence type you selected.
Check the sign behavior when d or r is negative.
Common sequence mistakes
Many wrong answers come from choosing the wrong sequence type. If the pattern adds a constant amount, it is arithmetic; if it multiplies by a constant factor, it is geometric. Another common mistake is forgetting that the formulas use n−1 in the exponent or step count because the first term already occupies position 1.
Do not swap difference and ratio.
Do not forget the n−1 adjustment.
Do not use the geometric sum formula when r = 1 without simplification.
Do not enter a non-integer term number.
Real-world applications
Arithmetic sequences appear in savings plans with fixed deposits, seat numbering, repeated additive changes, and stair-step pricing. Geometric sequences appear in population growth, depreciation, compound patterns, repeated scaling, and exponential models. The calculator helps because it lets you switch between the two ideas without rewriting the workflow from scratch.
Budgeting with fixed increases or decreases.
Repeated scaling in design and modeling.
Exponential growth and decay checks.
Classroom sequence and series practice.
When to use this calculator
Use this tool when you know the first term, know how the pattern repeats, and want a specific term or the total through a specific term. If you only need a named special sequence such as Fibonacci, a dedicated calculator may be better; this page is for the general arithmetic/geometric formulas that solve many routine problems quickly.
Use it for direct nth-term questions.
Use it when a finite sum is needed too.
Use it when a few preview terms help confirm the pattern.
Use a specialized tool if the pattern is not arithmetic or geometric.
Manual method versus calculator use
Listing terms by hand is perfect for spotting the pattern, but it becomes tedious once n grows. The calculator is most useful after you recognize the type and want the exact n-th term or partial sum fast. A strong habit is to estimate whether the result should be growing slowly, growing explosively, or decreasing before you trust the final output.
Write the first few terms by hand when learning.
Switch to the calculator for larger n or repeated comparisons.
Use the preview and explicit formula as self-checks.
Keep full precision when the ratio or difference is decimal-based.
Hızlı Referans Kartı
Sequence quick reference
Hızlı referans • Sequence Hesaplayıcı
Arithmetic: aₙ = a₁ + (n−1)d; Geometric: aₙ = a₁r^(n−1).Geçerli aralık: Use a finite first term, finite difference or ratio, and an integer term number from 1 to 500.
Yaygın Değerler
⚠ Dikkat
- •Do not treat a multiplicative pattern like an arithmetic sequence.
- •Do not enter a non-integer term number.
- •Do not forget the n−1 in the term formulas.
- •Watch for very large geometric ratios and term numbers that can overflow.
Pro İpuçları
- →Write the first few terms by hand to confirm the sequence type.
- →Use the preview as a quick reasonableness check.
- →If r = 1, the geometric sum becomes n times the first term.
- →A negative ratio means the signs will alternate.
SSS
What kinds of sequences does this calculator handle?
It handles arithmetic sequences and geometric sequences. The selector decides whether the step value is treated as a common difference or a common ratio.
What is the difference between arithmetic and geometric sequences?
Arithmetic sequences change by adding the same amount each step. Geometric sequences change by multiplying by the same ratio each step.
Why do the formulas use n minus 1?
Because the first term already occupies position 1. You need only n−1 additional differences or ratios to reach the n-th term.
What happens when the geometric ratio is 1?
Every term stays equal to the first term, so the sum of the first n terms simplifies to n × a₁.
Can the common difference or ratio be negative?
Yes. A negative difference creates a decreasing arithmetic sequence, and a negative ratio creates a geometric sequence with alternating signs.
Why does the calculator show the first five terms?
The preview helps you confirm that you selected the correct sequence type and entered the correct difference or ratio before relying on the closed-form result.
Do I have to enter a whole-number term number?
Yes. In this calculator, n must be a positive integer because it represents the position of a discrete term in the sequence.