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.
Sₙ = n/2 × (2a + (n−1)d) sums any arithmetic series without adding terms individually. Gauss reportedly discovered this at age 7 when asked to sum 1 + 2 + 3 + ... + 100: S₁₀₀ = 100/2 × (1+100) = 5,050. The formula works because the first and last terms always add to the same value (a + aₙ), and there are n/2 such pairs.
The nth term of an arithmetic sequence is aₙ = a + (n−1)d, where a is the first term and d is the common difference. For a = 3 and d = 5: the 20th term is 3 + 19 × 5 = 98. The mean of an arithmetic series is always (a + aₙ)/2, which equals the median — a property unique to evenly-spaced data. The Standard Deviation Calculator can verify this symmetry for any arithmetic data set.
Arithmetic series model linear accumulation: saving $200/month for 5 years with monthly $10 increases totals S₆₀ = 60/2 × (2×200 + 59×10) = 30 × 990 = $29,700. Stacking objects in triangular pyramids follows arithmetic sums: a layer with n items on each side has n(n+1)/2 objects, which is Sₙ for the series 1+2+3+...+n.
| Series | a | d | n | Sₙ |
|---|---|---|---|---|
| 1+2+3+...+100 | 1 | 1 | 100 | 5,050 |
| 2+5+8+...+29 | 2 | 3 | 10 | 155 |
| 10+7+4+...+−8 | 10 | −3 | 7 | 7 |
| 100+200+...+1000 | 100 | 100 | 10 | 5,500 |
A geometric series multiplies each term by a constant ratio r: Sₙ = a(1 − rⁿ)/(1 − r). For a = 1 and r = 2 over 10 terms: S₁₀ = 1 × (1 − 1024)/(1 − 2) = 1,023. The last term alone (512) is half the total sum — a hallmark of exponential growth where recent terms dominate.
When |r| < 1, the infinite sum converges: S∞ = a/(1−r). For a = 3 and r = 0.5: S∞ = 3/0.5 = 6. The partial sums approach 6 asymptotically: S₁ = 3, S₂ = 4.5, S₃ = 5.25, S₈ = 5.977. At r = 0.9, convergence is much slower: S∞ = 10, but S₁₀ = 6.513 and S₅₀ = 9.949. The closer |r| is to 1, the more terms are needed to approach the limit.
Geometric series underpin compound interest: a $1,000 investment at 5% annual return over 20 years grows via the geometric sum formula to $1,000 × (1.05)²⁰ = $2,653. Repeating decimal 0.333... = 3/10 + 3/100 + 3/1000 + ... is a geometric series with a=0.3 and r=0.1, converging to 0.3/0.9 = 1/3. The Fibonacci Calculator connects to geometric growth because the Fibonacci sequence grows at the rate of the golden ratio φ ≈ 1.618.
Sigma notation Σ compactly represents sums: Σ(i=1 to n) i = 1 + 2 + ... + n = n(n+1)/2. Common closed-form sums include Σ i² = n(n+1)(2n+1)/6 for the sum of squares and Σ i³ = [n(n+1)/2]² for the sum of cubes. The sum of the first 10 squares: 10 × 11 × 21 / 6 = 385.
Telescoping series simplify by cancellation. The sum Σ(1/k − 1/(k+1)) for k=1 to n collapses to 1 − 1/(n+1) because adjacent terms cancel. For n=100: sum = 1 − 1/101 = 100/101 ≈ 0.9901. Partial fractions decompose rational expressions into telescoping form: 1/(k(k+1)) = 1/k − 1/(k+1).
Harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges despite terms approaching zero. After 10 terms the sum is 2.93, after 100 it is 5.19, and after 10,000 it is 9.79. The rate of growth is approximately ln(n) + 0.5772 (Euler–Mascheroni constant). This slow divergence explains why equal-tempered musical tuning uses logarithmic frequency ratios. The Logarithm Calculator computes the ln(n) growth rate directly.
| Formula | Sum | Example (n=10) |
|---|---|---|
| Σ i | n(n+1)/2 | 55 |
| Σ i² | n(n+1)(2n+1)/6 | 385 |
| Σ i³ | [n(n+1)/2]² | 3,025 |
| Σ rᵏ (r=0.5) | a(1−rⁿ)/(1−r) | 1.998 |
Last Updated: Mar 26, 2026
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