What's the difference between arithmetic and geometric sequences?

An arithmetic sequence adds the same number each time (e.g., 2, 5, 8, 11 with difference +3). A geometric sequence multiplies by the same number each time (e.g., 2, 6, 18, 54 with ratio ×3).

Can I use this for Fibonacci sequences?

Yes. Select Fibonacci and enter your first term and second term. The calculator generates subsequent terms where each value is the sum of the two previous terms.

What does 'nth term' mean?

The nth term is the value at position n in your sequence. For example, the 5th term is the fifth number in the sequence.

Why would I need the sum of a sequence?

Summing sequences is useful in finance (calculating total payments), physics (distance traveled with constant acceleration), and planning (total resources needed over a period).

What if my common difference or ratio is negative?

That's fine. A negative difference in arithmetic sequences makes values decrease (e.g., 10, 7, 4, 1). A negative ratio in geometric sequences alternates signs (e.g., 2, −6, 18, −54).

Can the calculator handle decimal values?

Yes. You can enter decimals for the first term, common difference, ratio, or number of terms. The results will reflect your inputs precisely.

Number Sequence Calculator

Sequences explained

A number sequence is an ordered list of numbers following a specific rule. This calculator finds two key results: the nth term (the value at any position) and the sum (the total of all terms up to position n). It works with three common sequence types.

The formula

Arithmetic: nth term = a + (n−1)d | Sum = n/2 × (2a + (n−1)d)

Geometric: nth term = a × r^(n−1) | Sum = a × (1 − r^n) / (1 − r) (when r ≠ 1)

Fibonacci: nth term = sum of previous two terms | Sum = sum of all generated terms

Worked example

Scenario 1: Arithmetic sequence

You're saving money and deposit $50 in month 1, then $55 in month 2, $60 in month 3, and so on. What will you have deposited in month 12, and what's your total savings?

First term (a) = 50
Common difference (d) = 5
Number of terms (n) = 12

12th term: 50 + (12 − 1) × 5 = 50 + 55 = $105

Sum: 12/2 × (2 × 50 + (12 − 1) × 5) = 6 × (100 + 55) = 6 × 155 = $930

So in month 12 you deposit $105, and your total savings across all 12 months is $930.

Scenario 2: Geometric sequence

A viral video gets 1,000 views on day 1. Each day it gets 1.5× the previous day's views. How many views on day 8, and how many total views across 8 days?

First term (a) = 1,000
Common ratio (r) = 1.5
Number of terms (n) = 8

8th term: 1,000 × 1.5^(8−1) = 1,000 × 1.5^7 = 1,000 × 17.0859 = 17,085.9 views

Sum: 1,000 × (1 − 1.5^8) / (1 − 1.5) = 1,000 × (1 − 25.6289) / (−0.5) = 1,000 × (−24.6289) / (−0.5) = 49,257.8 total views

Scenario 3: Fibonacci sequence

You start a Fibonacci sequence with 1, 1. What are the first 10 terms, and their sum?

First term = 1
Second term = 1
Number of terms = 10

The sequence unfolds as: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

10th term: 55

Sum: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143

Common mistakes

Confusing position with value: The nth term is the position (like "month 5" or "day 10"), not the difference or ratio itself. Always use n to mean "which position?"

Forgetting the "minus 1" in formulas: Both arithmetic and geometric nth-term formulas use (n − 1), not n. This is because the first term is at position 1, not position 0.

Mixing up ratio and difference: Use a common difference for arithmetic (addition/subtraction) and a common ratio for geometric (multiplication/division). Entering 2 as a difference means +2 each step; entering 2 as a ratio means ×2 each step—very different results.

Assuming geometric sums work with r = 1: When the ratio equals 1, every term is identical, so the sum is simply a × n. The standard formula breaks down, so use this simpler version instead.

Sequences explained

The formula

Worked example

Common mistakes

Frequently asked questions

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