KEVOS® MAKEOS · Mathematics
Arithmetical and Geometrical Progressions
What is the last term and the sum of this arithmetical or geometrical progression?
Last term and sum of an arithmetical progression (constant difference) or a geometrical progression (constant ratio) from the first term, the difference or ratio and the number of terms, with the ratio or difference that fits a given first and last term.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Progression | choice | Choose the kind of progression. |
| First term | The first term. | |
| Common difference (arithmetical) | Difference between successive terms. | |
| Common ratio (geometrical) | Ratio of each term to the one before. | |
| Number of terms | Number of terms. | |
| Last term to fit (0 = not used) | Optional last term: the difference or ratio that reaches it in n terms is also reported. |
Method and formulas
- Arithmetical: l = a + (n − 1) d; S = n (a + l) ÷ 2.
- Geometrical: l = a rⁿ⁻¹; S = a (rⁿ − 1) ÷ (r − 1), or n a when r = 1.
- Fit: d = (l − a) ÷ (n − 1); r = (l ÷ a)^(1/(n−1)).
- Example: 8 terms from 2 with ratio 1.5 end at 34.172 and sum to 98.516.
Limitations
- Fitting a geometrical ratio needs first and last terms of the same sign.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Space a set of sizes or speeds between two limits
- Sum stepped quantities such as escalating costs
- Work out the ratio between steps of a series
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Last term
- Sum of all terms
- Difference or ratio that fits a given last term
Interpreting the result
- Machine speeds and sizes usually follow a geometrical progression; costs that rise by a fixed amount follow an arithmetical one.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 37–39. Topic reference; no source tables or recommended settings are reproduced.
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