For arithmetic and geometric sequences, identify the rule, count terms, then substitute into the formula. This guide connects a hand calculation to the calculator for students checking a finite n-th term and sum. All numerical examples are constructed teaching data.
Arithmetic or geometric: compare differences and ratios
| Values | Check | Choice |
|---|---|---|
| 7, 10, 13, 16 | Each difference is 3 | Arithmetic, d=3 |
| 1, 2, 4, 8 | Each ratio is 2 | Geometric, r=2 |
| 3, 3, 3, 3 | Difference 0 and ratio 1 | Either arithmetic d=0 or geometric r=1 |
| 1, 2, 4, 7 | Differences and ratios vary | Neither formula applies directly |
A ratio check cannot divide by a zero term. A few matching values do not uniquely determine all later terms; check that the problem actually specifies an arithmetic or geometric rule.
Check the conditions, then verify
Choose the rule before entering the values
For an arithmetic check, enter first term 7, difference 3 and count 10. Verify term 34 and sum 205 separately.
Formulas: an individual term is not the sum
Here a₁ is the first term, d the difference, r the ratio and n a positive integer count. aₙ is one term; Sₙ adds all terms from the first through the n-th.
| Type | Term aₙ | Finite sum Sₙ |
|---|---|---|
| Arithmetic | a₁+(n−1)d | n(a₁+aₙ)/2 |
| Geometric, r≠1 | a₁rⁿ⁻¹ | a₁(1−rⁿ)/(1−r) |
| Geometric, r=1 | a₁ | a₁n |
Starting at term 1 means nine changes before term 10. Adding an arithmetic sum forward and backward produces n copies of a₁+aₙ; dividing by two removes double counting. OpenStax explains these explicit-term and series formulas.
Worked examples: reproduce 205, 127 and 5.625
| Type | a₁ / d or r / n | Term | Sum |
|---|---|---|---|
| Arithmetic | 7 / 3 / 10 | 7+9×3=34 | 10×(7+34)/2=205 |
| Arithmetic | 1 / 1 / 100 | 100 | 100×101/2=5050 |
| Geometric | 1 / 2 / 7 | 2⁶=64 | (1−2⁷)/(1−2)=127 |
| Geometric | 3 / 0.5 / 4 | 0.375 | 3+1.5+0.75+0.375=5.625 |
| Geometric | 3 / 1 / 5 | 3 | 3×5=15 |
| Geometric | 2 / −1 / 4 | −2 | 2−2+2−2=0 |
Use the exponent guide to check the n−1 power. With a negative ratio, write out the first few terms to track alternating signs.
“Up to 50” versus “50 terms”: count before substituting
Starting at 7 and adding 3, values no greater than 50 satisfy 7+3(n−1)≤50. Thus n≤15.333…, giving 15 terms, last term 49 and sum 420. Entering n=50 answers a different question.
For d>0 and an upper limit L≥a₁, the count is floor((L−a₁)/d)+1. Negative differences, reversed inequalities or an initial term outside the bound require a fresh condition check.
Calculator steps and limits
- Open the Sequence Calculator and choose arithmetic or geometric. Complete inputs update results automatically.
- Enter the first term, difference or ratio, and count. Use 0.5 for a half; a fraction expression such as
1/2is not accepted. - Compare the term, sum and substituted working independently.
- Copy results with the sequence type, input values and formula into your own notes.
| Condition | Current behavior |
|---|---|
| Count n | Integer 1–10,000, with additional geometric integer limits below |
| Exact integer path | First term and difference/ratio are integers, each with magnitude at most 10⁹; BigInt is used |
| Geometric integer path | n≤500; when the ratio magnitude exceeds 10, n≤100 |
| Decimal or other path | Floating-point approximation; normally six decimal places, with scientific notation for small or large values |
| Integer input magnitude | Inputs exceeding 9,007,199,254,740,991 are rejected |
| Ratio 0 or numeric range failure | Error instead of adopting infinity or an unrepresentable answer |
Not every integer input uses BigInt. The supported integer example a₁=1, r=2, n=100 gives sum 1267650600228229401496703205375; commas in the display are grouping separators. See prime factorization for interpreting repeated factors as powers.
Finite and infinite sums have different conditions
This tool evaluates a finite number of terms. For a nonzero first term, the usual infinite geometric series converges to a₁/(1−r) when |r|<1. With a₁=3 and r=0.5, the infinite sum is 6, while the first four terms total 5.625. A finite r=2 sum of 127 does not imply an infinite sum exists.
Check the conditions, then verify
If results disagree, check the count and r=1 case
Match the original first term, rule and count. Favorite the tool for a future check after recording your working.
Update and verification scope
On October 27, 2026 the Tools Hub editorial team compared primary references with the formulas and input conditions in the public source. The examples were checked by hand and browser interaction. Processing time and exact decimal equality are not guaranteed. Favorites retain a tool shortcut in the same browser; they do not save or sync inputs or notes.
Find related lessons and tools in the mathematics category.
FAQ
Does “up to 50” mean 50 terms?
No. Starting at 7 with difference 3, terms no greater than 50 have count 15, last value 49 and sum 420. n is a count, so do not enter the bound 50 automatically.
What happens with ratios 1, negative values or 0?
For r=1, sum equals first term times count. A negative ratio alternates signs. This tool rejects r=0; writing a recurrence that multiplies by zero is separate from supported calculator inputs.
Can this calculator evaluate an infinite geometric series?
It handles finite n-term sums. For a nonzero first term, the usual infinite series converges to a₁/(1−r) when |r|<1. First term 3 and ratio 0.5 have infinite sum 6, but the first four terms total 5.625.
Is every integer calculation exact?
No. BigInt is used when first term and difference/ratio are integers with magnitudes at most 10⁹. Geometric integer calculations require n≤500, or n≤100 when the ratio magnitude exceeds 10. Other paths are approximate, with range errors for unsupported inputs.
