Sequences and Series Calculators
This category presents calculators describing numerical patterns and ordered progressions. Each tool provides outputs for arithmetic terms, geometric totals, recursive relationships and infinite series based on standard sequence structures.
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Arithmetic Sequence Calculator
An arithmetic sequence increases or decreases by the same fixed amount each time, creating a steady step-by-step pattern.
Example use: tracking the daily total of pages read when the number added each day stays constant.
Inputs: first term, common difference, term number
Outputs: common difference, value of the chosen term, sum of the first terms, arithmetic mean, summation function, sequence trend, crossing point
Visual: shows how each term grows alongside the running total
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Recursive Sequence Calculator
A recursive sequence builds each new term from earlier terms, producing patterns that may stabilise, grow steadily, or grow rapidly.
Example use: modelling how the number of steps taken on a walk increases when each day's total depends on the previous two days.
Inputs: preset sequence choice, first term, second term, first coefficient, second coefficient, constant term, number of terms
Outputs: number of terms calculated, last term value, sum of terms, convergence ratio, fixed point, growth classification, characteristic roots, dominant root
Visual: shows the term values, the running total, and the change between consecutive terms
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Geometric Sequence Calculator
A geometric sequence multiplies each term by the same constant factor, creating growth or decay that accelerates or slows in a predictable way.
Example use: tracking how many coins are saved when the amount set aside doubles each day.
Inputs: first term, common ratio, number of terms
Outputs: series status, sequence behaviour, value of the chosen term, sum of the first terms, product of the first terms, arithmetic mean, geometric mean, ratio percentage change, sum of reciprocals
Visual: shows the term values, the cumulative total, and the running mean
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Infinite Geometric Series Calculator
An infinite geometric series settles to a finite total whenever its terms shrink quickly enough from one step to the next.
Example use: estimating the total time spent on a task when each repeated attempt takes a smaller fraction of the previous one.
Inputs: first term, common ratio
Outputs: sum to infinity, convergence status, margin of error for twenty terms, percentage of total reached after twenty terms, terms needed to reach ninety-nine percent of the total
Visual: shows how the partial sums and individual terms change as more terms are added
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Telescoping Series Calculator
A telescoping series collapses many of its terms when expanded, leaving only a small number of values that determine its total.
Example use: summing a list of small adjustments where most increases and decreases cancel each other out.
Inputs: numerator value, difference value, starting index, upper limit
Outputs: partial sum, number of terms, series type, theoretical limit, tail remainder, convergence completion
Visual: shows the partial sums, individual terms, and how positive and negative parts contribute
Sequences and Series Calculators FAQs
An arithmetic sequence contains terms separated by a constant difference, forming a linear numerical progression across positions.
A geometric sequence contains terms generated by multiplying by a constant ratio, producing a multiplicative numerical pattern.
An infinite geometric series has a finite total when the ratio has an absolute value less than one, allowing the terms to converge to a numerical limit.
It identifies cancellation patterns within a sum, presenting a result based on the remaining initial and final terms after simplification.