AkCalculators

Sequence & Series Calculator — arithmetic and geometric sequences explained

Sequences and series are foundational concepts in algebra, calculus and discrete mathematics. A sequence is an ordered list of numbers (terms) defined by a rule; a series is the sum of terms of a sequence. Two especially common types are arithmetic sequences (constant difference between consecutive terms) and geometric sequences (constant ratio between consecutive terms). This Sequence & Series Calculator computes nth terms and sums for arithmetic and geometric sequences, and can also analyze a small list of sample terms to detect whether the sequence appears arithmetic, geometric, or neither.

Arithmetic sequences

An arithmetic sequence has the form:

a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ...

where a₁ is the first term and d is the common difference. The nth term is given by the simple linear formula:

a_n = a₁ + (n − 1)·d

The sum of the first n terms (the arithmetic series) can be computed directly using:

S_n = n/2 · (2a₁ + (n − 1)·d) = n · (a₁ + a_n)/2

These formulas are implemented in the calculator and the steps are shown so learners can follow the algebra and substitution.

Geometric sequences

A geometric sequence has the form:

a₁, a₁·r, a₁·r², a₁·r³, ...

where r is the common ratio. The nth term is:

a_n = a₁ · r^(n − 1)

The sum of the first n terms (for r ≠ 1) is given by:

S_n = a₁ · (r^n − 1) / (r − 1)

When r = 1, the sequence is constant and the sum simplifies to S_n = n·a₁. The calculator handles this special case explicitly and shows algebraic simplification in steps.

Custom sequence detection

In practice you might be given a few terms and asked to identify the sequence type. The Custom mode accepts 3–10 sample terms and applies simple numerical checks:

• Arithmetic test: Check whether the differences between consecutive terms are (approximately) equal: d_i = a_{i+1} − a_i. If differences are consistent (within a small tolerance when fractional values are involved), the sequence is arithmetic and d is reported.
• Geometric test: Check whether ratios between consecutive terms are consistent: r_i = a_{i+1} / a_i. If ratios are consistent and a_i ≠ 0, the sequence is geometric and r is reported.

If both tests fail, the tool reports "neither". The calculator prints the diagnostic differences and ratios so you can see exactly why a sequence is classified one way or another.

Using the calculator — examples

1. Arithmetic example: a₁ = 5, d = 3, n = 10. Then a₁₀ = 5 + (10 − 1)·3 = 32. Sum S₁₀ = 10/2 · (2·5 + 9·3) = 5 · (10 + 27) = 185.
2. Geometric example: a₁ = 2, r = 3, n = 6. Then a₆ = 2·3⁵ = 486 and S₆ = 2·(3⁶ − 1)/(3 − 1) = 2·(729 − 1)/2 = 728.
3. Custom detection: Terms 4, 7, 10, 13 → differences 3,3,3 → arithmetic with d=3. Terms 3, 6, 12 → ratios 2,2 → geometric with r=2.

Fractional and negative parameters

Both arithmetic and geometric formulas work with fractional and negative parameters. For example, an arithmetic sequence with d = −1/2 will decrease by 0.5 each step. A geometric sequence with r = 1/2 will shrink exponentially. The calculator accepts simple fractional input (like 3/4) and decimal input, showing exact substitution where possible and numeric evaluation with the chosen display precision.

Edge cases and special considerations

• Geometric r = 0: All terms after the first are zero; sums use the general formula but simplify accordingly.
• Geometric r = 1: Use S_n = n·a₁ to avoid division by zero in the standard sum formula.
• Custom detection with near-floating mismatch: When terms are computed from real-world measurements or decimals, small floating differences might make exact equality fail; the detection uses a numeric tolerance and prints diagnostics so you can adjust or interpret results.

Why this matters

Understanding sequences and series is crucial for calculus (limits and infinite series), finance (annuity sums), computer science (recurrence relations), and physics (discrete-time models). This calculator speeds computation and clarifies the algebraic steps so you can learn the reasoning behind results and check homework quickly.

Tips for students

• Always verify whether the problem expects integer-only sequences or allows fractional/real parameters.
• Use Custom mode to sanity-check sequences from problem statements — often sample terms reveal the rule directly.
• When computing large n with geometric sequences, be mindful of overflow and rounding when r>1; this tool gives numeric results but for symbolic proofs keep expressions in terms of r^n.

Conclusion

The Sequence & Series Calculator on AkCalculators is a practical, educational tool for computing nth terms and sums, exploring arithmetic and geometric patterns, and diagnosing sequence types from sample terms. Use the step-by-step mode to study the algebra or export results for reports and homework checks.