Quick answer
Find a₁ and d, then compute each term with aₖ = a₁ + (k - 1)d or by adding d repeatedly. Verify every neighbor difference.
Formula
- aₖ = a₁ + (k - 1)d
- Check: aₖ₊₁ - aₖ = d
Introduction
Listing terms is more than pushing buttons on a formula. You are proving that a story about equal steps is consistent from the first value through the last value you need.
The formula breakdown explains why (k - 1) appears; this article focuses on execution and checks.
For a guided tool walkthrough, see the calculator guide after you try one full example by hand.
What you need before you list
Without a₁ the list has no starting point. Without d the list has no repeatable step. Without a term count you do not know where to stop.
Some problems give a table instead of a story. Read the table in the order presented; the first row is not always a₁ unless the problem says so.
If differences between neighbors are not constant, stop and reconsider the model. Forcing an arithmetic template onto the wrong pattern produces confident wrong answers.
Apply the formula row by row
- aₖ = a₁ + (k - 1)d
- Alternate: add d to the previous term
For small n, repeated addition is often faster than rewriting the formula each time. For larger n, the formula prevents drift.
Label outputs as a₁, a₂, a₃ even if the problem uses plain language like "week 1, week 2." Matching notation reduces indexing errors.
When the problem asks only for the last term, you may stop after computing aₙ. Still verify d once so you know the model fits.
Step-by-step guide
- Identify a₁. Extract the first value from context, table, or diagram.
- Find d. Subtract consecutive terms at least twice to confirm constancy.
- Choose method. Use repeated addition for short lists; use the formula for distant indices.
- Compute each required term. Increase k by 1 and evaluate until you reach the required count.
- Verify. Confirm each neighbor difference equals d before you submit.
Full listing example
Given a₁ = 3 and d = 2, list five terms.
By addition: 3, 5, 7, 9, 11. By formula at k = 5: a₅ = 3 + 4×2 = 11.
Neighbor checks: 5 - 3 = 2, 7 - 5 = 2, and so on. The list is consistent.