Partial Sum Sₙ
100.0000
nth Term
19.0000
Mean
10.0000
| n | Term aₙ | Partial Sum Sₙ |
|---|---|---|
| 1 | 1.0000 | 1.0000 |
| 2 | 3.0000 | 4.0000 |
| 3 | 5.0000 | 9.0000 |
| 4 | 7.0000 | 16.0000 |
| 5 | 9.0000 | 25.0000 |
| 6 | 11.0000 | 36.0000 |
| 7 | 13.0000 | 49.0000 |
| 8 | 15.0000 | 64.0000 |
| 9 | 17.0000 | 81.0000 |
| 10 | 19.0000 | 100.0000 |
Sₙ = n/2 × (2a + (n−1)d)Calculates the sum of the first n terms of an arithmetic sequence.
Where:
Sₙ= Sum of the first n termsa= First term of the sequenced= Common difference between consecutive termsn= Number of terms to sumSₙ = a × (1 − rⁿ) / (1 − r)Calculates the sum of the first n terms of a geometric sequence when r ≠ 1.
Where:
Sₙ= Sum of the first n termsa= First term of the sequencer= Common ratio between consecutive termsn= Number of terms to sumS∞ = a / (1 − r), where |r| < 1The sum of an infinite geometric series converges to this value when the absolute value of the common ratio is less than 1.
Where:
S∞= Sum of the infinite seriesa= First term of the sequencer= Common ratio (must satisfy |r| < 1 for convergence)Inputs
Result
The arithmetic series 1, 3, 5, ..., 19 has 10 terms. Using Sₙ = n/2 × (a + aₙ) = 10/2 × (1 + 19) = 5 × 20 = 100. The mean is (1+19)/2 = 10.
Inputs
Result
With a=3 and r=0.5, S₈ = 3 × (1 − 0.5⁸) / (1 − 0.5) = 3 × 0.99609375 / 0.5 = 5.9766. The infinite sum converges to 3/(1−0.5) = 6.
Inputs
Result
A decreasing arithmetic series with a=5, d=−3: S₆ = 6/2 × (2×5 + 5×(−3)) = 3 × (10 − 15) = −15. The terms cross zero and become negative.
The sum of the first n terms of an arithmetic series is Sₙ = n/2 × (2a + (n−1)d), where a is the first term and d is the common difference. Equivalently, Sₙ = n/2 × (a + aₙ), where aₙ is the last term.
| Property | Arithmetic | Geometric |
|---|---|---|
| Pattern | a, a+d, a+2d, ... | a, ar, ar², ... |
| nth Term | a + (n−1)d | a × rⁿ⁻¹ |
| Partial Sum | n/2 × (2a + (n−1)d) | a(1−rⁿ)/(1−r) |
| Converges? | Never | Only if |r| < 1 |
A geometric series converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1). When it converges, the infinite sum is S∞ = a / (1 − r). If |r| ≥ 1, the series diverges.
| Ratio (r) | Behavior | Example (a=1) |
|---|---|---|
| r = 0.5 | Converges to 2 | 1 + 0.5 + 0.25 + ... = 2 |
| r = 0.9 | Converges to 10 | 1 + 0.9 + 0.81 + ... = 10 |
| r = 1 | Diverges (constant) | 1 + 1 + 1 + ... = ∞ |
| r = 2 | Diverges (grows) | 1 + 2 + 4 + ... = ∞ |
For an arithmetic sequence, the nth term is aₙ = a + (n−1) × d. For a geometric sequence, aₙ = a × rⁿ⁻¹. In both cases, a is the first term, d is the common difference, and r is the common ratio.
A sequence is an ordered list of numbers following a pattern (e.g., 1, 3, 5, 7). A series is the sum of a sequence's terms (e.g., 1 + 3 + 5 + 7 = 16). The partial sum Sₙ adds the first n terms of a sequence.
The mean (average) of an arithmetic series equals (a + aₙ) / 2, where a is the first term and aₙ is the last term. This works because arithmetic terms are evenly spaced. For example, the mean of 1, 3, 5, 7, 9 is (1+9)/2 = 5.
Last Updated: Mar 9, 2026
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