Quick answer
Examples show how sign and scale of d change the list while aₙ = a₁ + (n - 1)d stays the same.
Formula
- aₙ = a₁ + (n - 1)d
Introduction
Examples turn abstract rules into numbers you can defend on homework. The goal is not memorizing lists but seeing how one formula adapts to different signs and scales.
Review the formula article if notation feels shaky, then return here for numeric practice.
Before you find d from data, the finding common difference guide explains subtraction checks that every example below relies on.
Example types you should recognize
Positive d creates increasing lists. Negative d creates decreasing lists. Fractional d creates decimal steps that still satisfy constant differences.
Real-world wording may hide a₁ and d inside dollars, seats, or temperatures. Translate the story into numbers before you substitute.
Always end with a neighbor check. If a single subtraction fails, the entire arithmetic label is suspect.
Shared formula across cases
- aₙ = a₁ + (n - 1)d
- Verification: aₖ₊₁ - aₖ = d
Changing d or a₁ changes the list, not the structure of the formula. That is why practice with multiple signs matters.
When an example asks for many terms, list by addition for small n and by formula for the last term when n is large.
Match each example below in the calculator to build confidence that your hand work and the tool agree.
Step-by-step guide
- Classify the problem. Decide whether d should be positive, negative, or fractional.
- Write a₁ and d. Extract values from the prompt before you compute.
- Build the list. Use addition or substitution per term.
- Verify differences. Subtract neighbors to confirm d.
- Reflect. Note what changed between examples besides the numbers.
Three core patterns plus a context problem
Increasing: a₁ = 2, d = 3 gives 2, 5, 8, 11 for four terms. Check: each difference is 3.
Decreasing: a₁ = 10, d = -3 gives 10, 7, 4, 1. Check: each difference is -3.
Fractional: a₁ = 0.5, d = 0.5 gives 0.5, 1, 1.5, 2. Decimals do not break the rule.
Context: seats per row increase by 2 starting at 18 in row 1. Here a₁ = 18, d = 2. Row 5 has a₅ = 18 + 8 = 26 seats if the pattern stays arithmetic.