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Arithmetic progression sum and last term
What are the last term, sum and mean of n numbers starting at your first value and changing by the same difference each time?
Enter a first term, a constant signed difference and a whole number of terms. Increasing, decreasing and constant progressions are supported.
Your finite progression
The first term is position 1. Add d once for each following term; d can be negative or zero. Use a consistent numerical or unit basis. The count must be a positive whole number; a decimal is not entered as a/b.
How the estimate works
For positions k = 1 through n, t(k) = a + (k − 1) × d. The last term is a + (n − 1) × d. Pairing first and last gives S = n × (a + last) ÷ 2; the mean is (a + last) ÷ 2. There is no empty progression in this input model. With one term only the first value is needed for the results; input errors remain visible.
t(k) = a + (k − 1) × d; k = 1, …, n
last = a + (n − 1) × d
S = n × (a + last) ÷ 2
mean = (a + last) ÷ 2This is finite arithmetic with a constant additive difference. It does not infer a pattern from data, find missing inputs, handle changing differences or geometric progressions, compute infinite sums, forecast real values or list every term. Inputs are your chosen numbers, not recommended targets or a real-world guarantee.
OpenStax: arithmetic sequence definition and nth term ↗OpenStax: finite arithmetic series sum ↗
Check the filled example
Our example: a = 7, d = −1.5, n = 6. The terms are 7, 5.5, 4, 2.5, 1, −0.5. The last term is −0.5, their sum is 19.5 and their mean is 3.25. These are chosen mathematical inputs, not a forecast or a norm.
Accepted input values
- First term a
- 7
- Common difference d
- -1.5
- Number of terms n
- 6
Plan the next step
Use the standard calculator for separate operations on your numbers. This progression assumes a constant difference rather than checking a data pattern.
Open the standard calculator →