Sequences explained in plain English

A sequence is an ordered list of numbers that follows a rule. You’ll often see sequences written like: 2, 4, 6, 8, 10, … or 3, 6, 12, 24, …. The key word is ordered: the position of each number matters. We usually call the numbers in the list terms. The first term is written as a₁, the second as a₂, and so on. The n-th term (or aₙ) means “the term in position n.”

Sequence problems show up everywhere: saving money each month, predicting patterns, computing interest growth, analyzing data trends, and solving algebra homework. The most common “starter” sequences in math classes are arithmetic sequences (add the same amount each step) and geometric sequences (multiply by the same amount each step). This calculator focuses on those two because they cover a huge chunk of real-world and classroom sequence questions.

Knowing aₙ means you can jump directly to any position without listing every term. For example, instead of writing out 50 steps, you can compute the 50th term in one line using a formula. That’s why sequence formulas are so powerful: they turn “counting forward” into direct computation.

Arithmetic vs geometric (fast intuition)

• Arithmetic sequence: each term increases (or decreases) by the same amount d called the common difference.
• Geometric sequence: each term is multiplied by the same number r called the common ratio.

If you see a sequence where the difference between consecutive terms stays constant, it’s arithmetic. If the ratio (division) between consecutive terms stays constant, it’s geometric. When you’re unsure, compute a couple of differences and ratios. One of them will look “stable.”

• 2, 5, 8, 11… differences are +3 each time ⇒ arithmetic with d = 3.
• 3, 6, 12, 24… ratios are ×2 each time ⇒ geometric with r = 2.

Arithmetic sequence formulas (aₙ and Sₙ)

For an arithmetic sequence, you start at a₁ and add d each step. The n-th term formula is:

aₙ = a₁ + (n − 1)d

Why “(n − 1)”? Because to get from the first term to the n-th term, you take n − 1 jumps. Each jump adds d, so you add d that many times.

Sometimes you need the total of the first n terms, written as Sₙ. For arithmetic sequences, the sum is:

Sₙ = n/2 × (2a₁ + (n − 1)d)

You can remember this as “number of terms times average of first and last.” Because in an arithmetic sequence, terms pair up nicely: first + last, second + second-to-last, and so on.

Geometric sequence formulas (aₙ and Sₙ)

For a geometric sequence, you start at a₁ and multiply by r each step. The n-th term formula is:

aₙ = a₁ × r^(n − 1)

Again, it’s n − 1 multiplications to get from the first term to the n-th term. Every multiplication applies the same ratio r.

If r ≠ 1, the sum of the first n terms is:

Sₙ = a₁ × (1 − r^n) / (1 − r)

If r = 1, the sequence is constant (all terms equal a₁), so the sum is simply Sₙ = n × a₁. This calculator handles both cases automatically.

Examples you can copy (and screenshot)

Suppose a₁ = 7, d = 4, and you want a₁₀. Use aₙ = a₁ + (n − 1)d:

a₁₀ = 7 + (10 − 1)×4 = 7 + 36 = 43.

First 6 terms: 7, 11, 15, 19, 23, 27. Next terms follow the same +4 pattern.

Suppose a₁ = 3, r = 2, find a₈: a₈ = 3×2^(8−1) = 3×2^7 = 3×128 = 384.

First 6 terms: 3, 6, 12, 24, 48, 96. This is the classic “doubling” pattern.

If you save $10 in week 1 and increase savings by $5 each week, then a₁=10, d=5. Total after 12 weeks is: S₁₂ = 12/2 × (2×10 + (12−1)×5) = 6 × (20 + 55) = 6 × 75 = $450.

If a population starts at 100 and grows by 10% each period, then a₁=100 and r=1.1. The 6th term is a₆ = 100×1.1^5 ≈ 161.05. The total across 6 periods is computed with the geometric sum formula.

Behind the scenes (simple + reliable)

This Sequence Calculator asks for your sequence type, then uses the standard formulas above to compute: (1) the n-th term, (2) the first N terms for a quick view, and (3) the sum of the first n terms. Everything runs locally in your browser—no signup and no server calls—so it loads instantly and is easy to share.

• n must be a positive integer (1, 2, 3…). If you type 0 or a decimal, the calculator will ask you to fix it.
• Use decimals if needed (e.g., r = 1.05 for 5% growth). The calculator will display rounded outputs for readability.
• Negative d or r are allowed. A negative ratio makes the sequence alternate signs (common in some algebra problems).

Pro tip for virality: compute a few terms, take a screenshot, and caption it “My brain likes patterns” or “Math rizz: unlocked.” 🙂

Sequence Calculator FAQs

• What is the difference between a sequence and a series?

  A sequence is the list of terms (a₁, a₂, a₃…). A series is the sum of those terms (Sₙ). This calculator shows both when relevant.

• How do I know if a sequence is arithmetic or geometric?

  Check consecutive differences for arithmetic and consecutive ratios for geometric. If one stays constant (or clearly repeats), that’s your type.

• Can d or r be negative?

  Yes. Negative d means the sequence decreases. Negative r makes the sequence alternate positive/negative values.

• What if r = 1?

  Then every term equals a₁. The sum becomes Sₙ = n×a₁. The calculator handles this special case automatically.

• Why does the formula use (n − 1)?

  Because you take (n − 1) steps from the 1st term to the n-th term. It’s “number of jumps,” not “number of terms.”

• Does this work for Fibonacci or custom rules?

  This page focuses on arithmetic and geometric sequences. For custom rules (like Fibonacci), you’d need a different recurrence-based calculator.

• Is this calculator accurate?

  Yes for arithmetic and geometric sequences. For very large n with large ratios, results can get huge and may display in scientific notation.

• Can I use decimals?

  Absolutely. Decimals are common for growth rates (like 1.08) or steady changes (like d = 0.25).

• What’s the most common real-world use?

  Arithmetic sequences model linear growth (adding the same amount each period). Geometric sequences model exponential growth (multiplying by a rate).

• Do you store my numbers?

  No. Everything runs locally. If you click “Save Result,” it stores the result only in your browser’s local storage on this device.

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Accuracy + responsibility

This calculator is designed for math learning, homework checking, and fast pattern exploration. If you’re using sequences for financial decisions, double-check inputs and rounding. Small changes in a geometric ratio can cause big differences over time.