Sequences & Series Calculator

📋 How to use this calculator

1. Pick the tab that fits your task: nth Term Arithmetic, Sum of n Arithmetic Terms, nth Term Geometric, Geometric Sum, Infinite Geometric Series, or Pattern Detection.
2. For arithmetic, enter the first term a, common difference d (the gap between terms), and term index n; the tool returns U_n and a step-by-step solution.
3. For an arithmetic sum, enter a, d, and the number of terms n; the output is S_n shown both ways (S_n = n/2*(2a+(n-1)d) and S_n = n/2*(a+U_n)).
4. For geometric, enter a, common ratio r, and n; r must be nonzero; the output is U_n = a*r^(n-1).
5. For an infinite geometric series, enter a and r; the series converges only when |r| < 1, otherwise it diverges and has no finite sum.
6. For pattern detection, enter the first 4-6 terms (comma-separated); the tool checks whether the sequence is arithmetic (constant difference) or geometric (constant ratio) and projects the next terms.

🧮 Arithmetic and geometric sequences and series

• a = U_1 = first term
• d = common difference = U_n - U_(n-1)
• r = common ratio = U_n / U_(n-1)
• n = term index (n = 1, 2, 3, ...)
• S_n = sum of the first n terms
• S_inf = sum of an infinite geometric series (converges only when -1 < r < 1)

For r = 1 in a geometric series, S_n = n*a (every term equals a). An arithmetic series diverges as n approaches infinity (unless a = 0 and d = 0). A geometric series converges to a/(1-r) only when |r| < 1.

💡 Worked example: Example: simple-interest savings (arithmetic sequence)

Result: End-of-month-12 balance is $1,600 with total interest of $600 (a hypothetical 60% annual return, not realistic in practice).

❓ Frequently asked questions

📚 Sources & references

• Sequences and Series - OpenStax College Algebra — OpenStax / Rice University
• Arithmetic Progression - Wolfram MathWorld — Wolfram Research
• Geometric Series - Wolfram MathWorld — Wolfram Research