Sequence & Series Calculator Online - Arithmetic & Geometric

What is a Sequence & Series Calculator?

A Sequence & Series Calculator is a digital tool designed specifically for calculating sequences and series in mathematics - number patterns that follow specific rules. A sequence is an ordered arrangement of numbers following a particular pattern (terms), while a series is the sum of the terms of a sequence. This calculator is very useful for middle and high school students studying sequences and series in mathematics. This topic frequently appears in national exams and college entrance tests. Additionally, the concepts of sequences and series are widely used in compound interest calculations, population growth, and loan installment calculations. The KalkuLab Sequence & Series Calculator provides various calculation modes: Arithmetic Sequence (nth term), Arithmetic Series (sum of n terms), Geometric Sequence (nth term), Geometric Series (sum of n terms), and Infinite Geometric Series.

Sequence & Series Formulas

Arithmetic: Un = a + (n-1)b, Sn = n/2(a + Un). Geometric: Un = a × rⁿ⁻¹, Sn = a(rⁿ-1)/(r-1)Formula: Infinite Geometric Series: S∞ = a/(1-r) for |r| < 1

Variables:

Unnth Term (Arithmetic)
The value of the term at position n in an arithmetic sequence(e.g.: U₁₀ = 3 + 9(2) = 21)
💡 Finding the value of a specific term
SnSum of n Terms (Series)
The sum from the first term to the nth term(e.g.: S₁₀ = 10/2(3+21) = 120)
💡 Calculating the total sum of numbers
a / U₁First Term
The first number in the sequence(e.g.: a = 3)
💡 Starting point of the sequence
dCommon Difference (Arithmetic)
The constant difference between consecutive terms(e.g.: d = 2 (sequence: 3,5,7,9,...))
💡 Determining the pattern of increase/decrease
rCommon Ratio (Geometric)
The constant ratio between consecutive terms(e.g.: r = 2 (sequence: 3,6,12,24,...))
💡 Determining the multiplication factor
nNumber of Terms / Position
The number of terms being calculated(e.g.: n = 10 (first 10 terms))
💡 Determining the calculation limit

Categories:

Arithmetic SequenceUn = a + (n-1)d

Arithmetic SeriesSn = n/2(a + Un)

Geometric SequenceUn = a × rⁿ⁻¹

Geometric SeriesSn = a(rⁿ-1)/(r-1)

Infinite Geometric SeriesS∞ = a/(1-r), |r|<1

How to Use the Sequence & Series Calculator on KalkuLab

Using the KalkuLab Sequence & Series Calculator is very easy. Follow these simple steps:

1
Select Calculation Mode
Choose the calculation type: Arithmetic Sequence (nth term), Arithmetic Series (sum of n terms), Geometric Sequence (nth term), Geometric Series (sum of n terms), or Infinite Geometric Series.
2
Enter Known Values
Enter the known values: a (first term), d (common difference - for arithmetic), r (common ratio - for geometric), n (number of terms), or Un (nth term).
3
Press the Calculate Button
Press the 'Calculate' button to process the calculation. The result will display the sought value along with step-by-step solutions.
4
View Results and Explanation
The calculation results will be displayed along with explanations of how it is calculated. You can see the formulas used and the steps involved.
5
Use the Reset Feature
Press the 'Reset' button to calculate another sequence/series. You can calculate various number patterns in sequence.

💡 Tip:

•Make sure to select the correct mode (arithmetic vs geometric)
•Common difference (d) can be positive (increasing) or negative (decreasing)
•Geometric ratio (r): |r| < 1 means approaching 0, |r| > 1 means growing larger
•Infinite geometric series is only valid if |r| < 1 (r ≠ 1)
•Use a dot (.) for decimal numbers, e.g., 2.5 for 2 1/2

Examples

Example 1: Saving Money Each Month (Arithmetic)

Problem:

A person saves $100 in the first month and increases savings by $20 each subsequent month. How much total savings after 12 months?

Solution:

1.Mode: Arithmetic Series (sum of n terms)
2.a = 100, d = 20, n = 12
3.Sn = n/2(2a + (n-1)d)
4.S₁₂ = 12/2(2×100 + 11×20) = 6(200 + 220) = 6×420 = $2,520

Result:Total savings after 12 months = $2,520

Using arithmetic series, we can calculate the total amount saved when the savings increase by a fixed amount each month.

Example 2: Population Growth (Geometric)

Problem:

A city has a population of 1 million people. If the population grows by 3% each year (r=1.03), what will be the population after 10 years?

Solution:

1.Mode: Geometric Sequence (nth term)
2.a = 1000000, r = 1.03, n = 11 (year 10 = 11th term if year 0 is term 1)
3.Un = a × rⁿ⁻¹
4.U₁₁ = 1000000 × 1.03¹⁰ ≈ 1000000 × 1.3439 ≈ 1,343,900

Result:Population after 10 years ≈ 1.34 million people

Population growth follows a geometric (exponential) pattern. In 10 years, the population grew by approximately 34%.

Frequently Asked Questions

What is the difference between a sequence and a series in mathematics?

A sequence is an ordered list of numbers following a specific pattern (terms: U₁, U₂, U₃,...). A series is the sum of the terms of a sequence (S = U₁ + U₂ + U₃ + ...). Example: Sequence: 2,4,6,8,... Series: 2+4+6+8+... = Sₙ.

How do you distinguish between arithmetic and geometric sequences?

Arithmetic: the difference between consecutive terms is constant (common difference d). Example: 3,5,7,9,... (d=2). Geometric: the ratio between consecutive terms is constant (common ratio r). Example: 3,6,12,24,... (r=2). Key characteristic: arithmetic uses +/–, geometric uses ×/÷.

When do you use an infinite geometric series?

Use an infinite geometric series (S∞ = a/(1-r)) only when |r| < 1 (ratio between -1 and 1). This means the terms get progressively smaller approaching 0. Example: a=10, r=0.5 → S∞ = 10/(1-0.5) = 20. If |r| ≥ 1, the series does not converge (S∞ is infinite).

Can an arithmetic sequence have a negative common difference?

Yes, the common difference (d) can be negative, meaning the sequence is decreasing. Example: 20,17,14,11,... (d = -3). The formula remains the same: Un = a + (n-1)d. For negative d, the term values get smaller as n increases.

What is the middle term in an arithmetic series?

The middle term in an arithmetic series with an even number of terms is the average of the two middle terms. For odd n, the middle term is the (n+1)/2 term. Example: series 2,4,6,8,10 (n=5), middle term = U₃ = 6. Property: Sₙ = n × middle term (for odd n).

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