BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Progression
- What it is
- The kind of sequence: arithmetical, where each term adds a fixed difference to the one before, or geometrical, where each term multiplies the one before by a fixed ratio.
- Units and format
- Choose one of 2 options from the list: Arithmetical (constant difference); Geometrical (constant ratio).
- Example
- The form opens with “Geometrical (constant ratio)” selected as a worked example. Choose the option that matches your case.
- Get it right
- The arithmetical option uses the common difference; the geometrical option uses the common ratio. The other is ignored.
What this means: First term
- What it is
- The first term of the sequence, the value the progression starts from, such as the smallest size or first payment.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- To fit a ratio to a last term, the first and last terms must have the same sign.
What this means: Common difference (arithmetical)
- What it is
- The fixed amount added to each term to get the next one in an arithmetical progression. A negative difference makes the terms fall.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The last term rises by the number of terms less one for each unit of difference.
- Get it right
- Used only for the arithmetical option.
What this means: Common ratio (geometrical)
- What it is
- The fixed multiplier from each term to the next in a geometrical progression: the second term divided by the first.
- Units and format
- Enter a plain number with no unit. The form accepts -1000000 to 1000000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only for the geometrical option. A ratio of 1 gives equal terms; between 0 and 1 the terms shrink.
What this means: Number of terms
- What it is
- How many terms the progression has, counting the first and the last.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Count the terms, not the gaps between them: five sizes have four steps.
What this means: Last term to fit (0 = not used)
- What it is
- An optional last term you want the progression to reach after the given number of terms. The tool then reports the difference or ratio that gets there.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A value of 0 means not used, so a last term of exactly zero cannot be fitted.
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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Open the Arithmetical and Geometrical Progressions calculator