How this calculation works
An arithmetic sequence adds the same difference at every step. The sum is the number of terms multiplied by the average of the first and last terms.
Nth term = first + (n − 1) × difference; sum = n × (first + last) ÷ 2A worked example
Starting at 3 and adding 2 gives a 10th term of 21 and a sum of 120.
Good to know
Uses the stated formula and assumptions. Displayed decimal values are rounded; use consistent units for all inputs.
Pairing the ends explains the sum
In the sequence 3, 5, 7, …, 21, the first and last terms add to 24. So do the second and next-to-last terms. Ten terms form five such pairs, giving a total of 120. The sum formula is a compact way to express that pairing.
A constant difference means growth is additive. Every step contributes the same increment; doubling the term number does not generally double the term because the starting value also matters.
Watch the gap between the first and nth term
There are n − 1 steps from the first term to the nth term. Using n steps is a common off-by-one error. For a sequence beginning at 3 with difference 2, the first term is 3, not 5.
A negative common difference produces a decreasing sequence. The formula still applies, and the sum can be positive, zero or negative depending on the terms included.
A few questions, answered
What does n = 1 return?
The first term and a sum equal to that first term. No differences have been added yet.
Can the difference be zero?
Yes. Every term is the same, and the sum is n times the first term.