Library / Foundations: Maths / FMA-37
Foundations: Maths
Geometric series sum
Sum of n terms where each term is r times the one before.
- Ref
- FMA-37
- Rev
- v1
- Field
- Foundations: Maths
- Result
- S
a is the first term and r the common ratio, so the terms are a, ar, ar², … and n is how many are added. For |r| < 1 the sum approaches a/(1 − r) as n grows. When r = 1 the formula is replaced by S = na.
Sum of n terms where each term is r times the one before. If r is below 1 the sum settles towards a limit; above 1 it grows without bound.
Calculator
- Last term a·rⁿ⁻¹512
- Last term as a share of the sum50.05 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Doubling ten times from 1 gives 1 + 2 + 4 + … + 512 = 1023, one less than the next term, 1024: each term is larger than all the earlier ones together.
- Compound growth with a rate i per period is a geometric series with r = 1 + i; the same formula totals regular payments or repeated-decay effects.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Number of terms | 1 to 40 | 1 | 1.1e12 | 1.1e12 | +1 | +100 % |
| a First term | 0.01 to 100 | 10.23 | 102,300 | 102,300 | +10% | +10 % |
| r Common ratio | 0.1 to 3 | 1.111 | 29,520 | 29,520 | +10% | +116 % |
Sweep of Common ratio
| Common ratio | Sum of terms |
|---|---|
| 0.1 | 1.111 |
| 0.4625 | 1.86 |
| 0.825 | 4.88 |
| 1.188 | 24.41 |
| 1.55 | 143.7 |
| 1.913 | 716.3 |
| 2.275 | 2,912 |
| 2.638 | 9,948 |
| 3 | 29,520 |
Sweep one variable
Variables and how each one contributes
Sum of terms
What the equation gives you. Everything else feeds it.
First term
The first term of the series. Scales every term, so the sum is directly proportional to a.
Common ratio
The factor each term is multiplied by to give the next. Very strong when r is above 1: a 10 % rise in r from 2 to 2.2 raises the sum of ten terms by about 116 %. For r below 1 it matters much less.
Number of terms
The number of terms added. With r above 1 each extra term roughly doubles the sum again (at r = 2); with r below 1 extra terms add less and less.
Worked example with the values above
- Ratio to the power n: rⁿ = 210 = 1,024.
- Numerator: |1 − rⁿ| = 1,023; denominator: |1 − r| = 1. Both are negative when r is above 1, so the signs cancel.
- Sn = 1 × 1,023 / 1 = 1,023.
Assumptions and limits
- A constant ratio between neighbouring terms.
- n is a positive whole number; r ≠ 1 in the formula (r = 1 gives na).
Common mistakes
- Using n instead of rⁿ in the numerator.
- Using the sum-to-infinity formula when r ≥ 1, where it does not apply.
Statistics and uncertainty
For r above 1 small errors in r are heavily amplified: at the defaults a 1 % error in r gives about an 8 % error in the sum. For r below 1 the sum is insensitive to r after the first few terms.
Where it is used
- Totals under repeated growth or decay, such as compound interest or repeated halving.
- Counting the nodes in a branching tree.
Maths you need
- Geometric sequences: each term is r times the one before
- Indices: rⁿ