Number Sequence Calculator
Identify and continue arithmetic and geometric sequences. Find the nth term and sum of the first n terms.
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How to use this calculator
Arithmetic sequences add a constant difference d; geometric sequences multiply by a constant ratio r.
- 1
Enter the first and second terms of your sequence.
- 2
Enter n to find the nth term of the sequence.
- 3
Enter N to find the sum of the first N terms.
- 4
Both arithmetic (constant difference) and geometric (constant ratio) results are shown.
Frequently asked questions
Why does it show both an arithmetic and a geometric answer?
Two terms cannot tell you which pattern you mean. 3 and 7 fit an arithmetic sequence with difference 4 and a geometric one with ratio 2.3333, so the calculator gives both, labelled separately. Pick the block that matches your sequence; for 3, 7, 11, 15 that is the arithmetic one.
What does the calculator do with a first term of 0?
The arithmetic results still appear: first term 0 and second term 5 give a difference of 5, a 5th term of 20 and a sum of 50 over 5 terms. The geometric block is left out because the ratio would need a division by zero. A second term of 0 also removes the geometric block, since every later term would be 0.
What happens when both terms are equal?
The difference is 0 and the ratio is 1. The arithmetic block reports the constant: first term 3 and second term 3 give a 4th term of 3 and a sum of 12. The geometric block is skipped when the ratio is exactly 1 because it would only repeat that.
What values can n and N take?
Both must be whole numbers of 1 or more. Fractional positions such as 2.5, zero and negative numbers return an error, and clearing either box counts as zero. n is the position of the term to find, and N is how many terms to add from the start.
When is the infinite sum shown as a number?
Only when the geometric ratio has an absolute value below 1. Terms 1 and 0.5 give a ratio of 0.5 and an infinite sum of 2. For a ratio of 1 or more in size, the output prints ∞, because the partial sums do not settle on a limit.
How does a negative ratio behave?
The terms alternate in sign. Starting 3, −6 gives a ratio of −2: the 5th term is 48 and the sum of the first 5 terms is 33. The ratio is rounded to four decimals in the label only; calculations use the unrounded value.
Can it find a rule for Fibonacci or square numbers?
No. It only models constant-difference and constant-ratio patterns from the first two terms. Sequences such as 1, 4, 9, 16 or 1, 1, 2, 3, 5 fall into neither group, and the first two terms of either would yield an arithmetic or geometric answer that does not match the real sequence.
Will it list the terms of the sequence?
No, it returns the nth term and the sum of the first N terms for each pattern. To see the individual terms, apply the difference or ratio repeatedly, or run the calculator for n = 3, n = 4 and so on.
nth terms and sums for arithmetic and geometric sequences
How two terms define the pattern
From a₁ and a₂ the calculator computes a difference d = a₂ − a₁ and a ratio r = a₂ ÷ a₁. The arithmetic nth term is a₁ + (n − 1)d and the sum of N terms is N ÷ 2 × (2a₁ + (N − 1)d). The geometric nth term is a₁ × rⁿ⁻¹ and its finite sum is a₁(1 − rᴺ) ÷ (1 − r). Both are produced every time because the first two terms fit either family; you choose the one that matches the rule you were given.
| a₁, a₂ | n = N | Arithmetic term / sum | Geometric term / sum |
|---|---|---|---|
| 3, 7 | 10 | 39 / 210 | 6150.526901 / 10761.172077 |
| 2, 6 | 6 | 22 / 72 | 486 / 728 |
| 1, 0.5 | 5 | −1 / 0 | 0.0625 / 1.9375 |
| 1000, 1080 | 10 | 1720 / 13600 | 1999.004627 / 14486.562466 |
| 3, −6 | 5 | −33 / −75 | 48 / 33 |
Worked example: 2, 6
With a₁ = 2 and a₂ = 6, n = 6 and N = 6, the arithmetic reading has d = 4 and gives 22 for the 6th term, with the first six terms adding to 72. The geometric reading has r = 3 and gives 486 for the 6th term and a sum of 728. The gap between 72 and 728 shows how fast a ratio overtakes a difference. If your list were 2, 6, 10, 14, you would use the arithmetic block; if it were 2, 6, 18, 54, the geometric one.
Worked example: a deposit growing by 8%
Entering 1000 and 1080 describes a balance growing by 8% each period. The geometric 10th term is 1999.004627, meaning the balance after nine periods of growth is about 1,999.00, and the ten-term total is 14486.562466. The arithmetic reading of the same two numbers is a flat 80 added each period, reaching 1720 at the 10th term. The difference of roughly 279 between the two 10th-term figures is the effect of compounding.
Convergence and alternating signs
Starting from 1 and 0.5, the ratio is 0.5, the 5th term is 0.0625, the first five terms add to 1.9375, and the infinite sum is 2. Each extra term closes part of the remaining gap to 2. A negative ratio flips the sign each step: 3 and −6 give a 5th term of 48 and a sum of 33 over five terms. Such a sequence never settles because |r| is 2, so no infinite sum is offered.
What the calculator leaves out
It does not guess a rule from three or more terms, so it cannot recognise Fibonacci, squares or any pattern beyond constant difference and constant ratio. Positions must be whole numbers from 1 upward. If your sequence has a known rule outside these two families, evaluate that rule directly rather than relying on the first two terms.
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