Salary progression: starts at ₹6 lakh, ₹50k annual increment, over 10 years
AP with a₁ = 6, d = 0.5 (lakh). a₁₀ = 6 + 9 × 0.5 = 10.5 lakh. Total earnings: S₁₀ = 10/2 × (6 + 10.5) = 82.5 lakh over 10 years. Classic AP-modelled career path.
Number Sequence Calculator - 無料オンライン電卓
Generate arithmetic, geometric and Fibonacci sequences. See the first n terms, the sum of the series, and the nth-term formula.
- 計算式をページに表示
- ブラウザ内で動作
- 入力内容は非公開
Last term aₙ
29
Sum Sₙ
155
Formula
aₙ = a + (n−1)d
= 2 + (n−1)·3
Sequence
2, 5, 8, 11, 14, 17, 20, 23, 26, 29
一部のコンテンツはまだ英語で表示されます。
Number Sequence Calculatorとは
Sequences and series are a core Class 10-11 mathematics topic in India, with arithmetic progressions (AP) and geometric progressions (GP) appearing in board exams and competitive tests like JEE Main, JEE Advanced and NEET. Beyond schoolwork, these patterns appear constantly in real life: salary increments are arithmetic (₹5 lakh starting + ₹50,000 each year), compound interest is geometric (₹1 lakh × 1.08 each year), EMI schedules are geometric, population growth is geometric, and Fibonacci numbers appear in nature (sunflower seed spirals, pine cone scales, flower petals).
This calculator generates the first n terms and the cumulative sum for three sequence types: arithmetic (constant difference), geometric (constant ratio), and Fibonacci (each term equals sum of previous two). The formula for nth term and series sum is displayed alongside the result so students can learn the math, not just get answers. AP and GP problems are JEE staples - sum to n terms, finding missing terms, applying AM-GM inequality, and solving word problems all rely on understanding these patterns deeply.
Number Sequence Calculatorの使い方（4ステップ）
Pick a sequence type: arithmetic, geometric, or Fibonacci.
Enter the first term and common difference / ratio (where applicable).
Choose how many terms to generate.
Read the full sequence and the sum.
Number Sequence Calculatorのしくみ
Arithmetic adds a constant difference each term: a, a+d, a+2d, … Geometric multiplies by a constant ratio: a, ar, ar², … Fibonacci adds the previous two terms: 0, 1, 1, 2, 3, 5, …
Formula
Arithmetic: aₙ = a + (n−1)d, Sₙ = n/2 [2a + (n−1)d] Geometric: aₙ = a·r^(n−1), Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1 Fibonacci: F(n) = F(n−1) + F(n−2)
- aFirst term
- d / rCommon difference / ratio
- nTerm index
Number Sequence Calculatorが結果を出す方法
Arithmetic Progression (AP): each term differs from the previous by a constant 'd' (common difference). aₙ = a₁ + (n-1)d. Sum of first n terms: Sₙ = n/2 × [2a₁ + (n-1)d] = n/2 × (first + last). AP appears in salary increment patterns, evenly-spaced time intervals, simple interest accruals, and basic counting problems.
Geometric Progression (GP): each term equals the previous multiplied by constant 'r' (common ratio). aₙ = a₁ × r^(n-1). Sum: Sₙ = a₁ × (r^n - 1)/(r-1) for r ≠ 1. Sum of infinite series: S∞ = a₁ / (1-r) for |r| < 1 (converges). GP is the math of compound interest, exponential decay, population growth, and most natural growth/shrinkage processes.
Fibonacci sequence: F(n) = F(n-1) + F(n-2) with F(0) = 0, F(1) = 1. Each term is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... The ratio F(n+1)/F(n) approaches the golden ratio φ = (1+√5)/2 ≈ 1.618 as n grows. Fibonacci numbers appear in plant phyllotaxis, breeding population models (Fibonacci's original rabbit problem), Pingala's metres in Sanskrit prosody, and financial chart analysis (Fibonacci retracements).
計算例
Compound interest: ₹1 lakh at 10% for 10 years
GP with a₁ = 1 lakh, r = 1.1. a₁₀ = 1 × 1.1⁹ = 2.358 lakh. After 10 years, principal compounds to 2.594 lakh (with a₁₀ being end-of-year-10 balance for ratio interpretation).
Fibonacci sequence: first 15 terms
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Notice the ratio between consecutive terms approaching φ: 377/233 ≈ 1.6180. Classic illustration of golden ratio convergence.
Number Sequence Calculatorを使う場面
- Class 10-11 mathematics - arithmetic and geometric progressions chapter.
- JEE Main, JEE Advanced - sequence and series questions.
- Financial planning - compound interest, EMI schedules, depreciation.
- Population modelling - exponential growth.
- Programming algorithms - recursive Fibonacci, dynamic programming.
- Statistics - geometric mean computation.
- Music theory - frequency ratios in scales.
- Nature studies - phyllotaxis, biological growth patterns.
よくある間違いと落とし穴
- Confusing AP and GP. AP adds; GP multiplies.
- Forgetting n-1 in formula. nth term uses (n-1), not n.
- Treating infinite geometric series as converging when |r| ≥ 1. It diverges.
- Computing Fibonacci naively with recursion - exponential time. Use iteration or matrix exponentiation for large n.
- Mixing first-term conventions. Some sources count from F(0) = 0, others from F(1) = 1.
このツールを使いこなす
- For salary growth and increments, use AP (arithmetic). For investment returns and compounding, use GP (geometric).
- Sum of an AP: average of first and last term, times count. Easy mental shortcut.
- Sum of GP: useful for compound interest and EMI total interest calculations.
- Golden ratio φ = 1.618 appears in Fibonacci ratios, design, architecture - has aesthetic appeal.
- Pair with the SIP, compound interest, EMI, and CAGR calculators on this site for related real-world sequences.
Number Sequence Calculatorに関するよくある質問
What's the difference between arithmetic and geometric?
Arithmetic adds a constant; geometric multiplies by a constant. Salary with fixed annual increment is arithmetic. Compound interest is geometric.
How fast does a Fibonacci sequence grow?
Roughly exponentially with ratio φ ≈ 1.618 (the golden ratio). F(50) is already over 12 billion, F(100) is 21 digits long.
Sum of an infinite geometric series?
Converges to a / (1 − r) when |r| < 1. Diverges (becomes infinite) when |r| ≥ 1. Useful for repeating decimals and economic multipliers.
Where do Fibonacci numbers appear?
Plant phyllotaxis (sunflower seeds, pine cones), animal breeding rates (Fibonacci's original rabbit problem), financial wave analysis, and golden-ratio aesthetics.
Can I generate harmonic or quadratic sequences?
Not directly here - these are common but less standard. The custom expression evaluator (Scientific Calculator) handles single terms; sequences need their own tool.
What is the nth term formula?
A closed-form expression for the nth term without computing earlier terms. Arithmetic: aₙ = a + (n−1)d. Geometric: aₙ = a·r^(n−1). Fibonacci has the Binet formula but it's irrational.
Math in your browser
From simple percentages to scientific notation, statistics, geometry and big-number arithmetic - every common math problem you'd reach for a calculator for, runs in your browser with no installs. The fraction calculator preserves exact rationals, the big-number calculator uses BigInt for hundreds of digits without precision loss, and the scientific calculator's expression evaluator is a hand-written shunting-yard parser (no eval) that supports trig, log, factorial and parentheses.
If you're a student, double-check your work without paying for a graphing calculator. If you're a professional, use these tools as a sanity check on spreadsheet formulas. Every result is shown alongside its formula and inputs, so the math is always auditable.
The Number Sequence Calculator is part of the Math suite on Online Calculator Tools, alongside 6 related calculators. All tools share the same UI patterns, the same privacy-first rendering (nothing is sent to a server), and the same commitment to showing the formula and assumptions behind every result.