Arithmetic nth Term
The nth term of an arithmetic sequence.
Formula
Live Calculator
MembersVariables
Example
Sequence 3, 7, 11, … (a₁=3, d=4): the 10th term is a₁₀ = 3 + 9×4 = 39.
How it works
In an arithmetic sequence, each term changes by the same fixed amount, called the common difference. Starting from the first term, you can find the nth term by adding that difference exactly n-1 times, because the first term needs 0 additions, the second needs 1, and so on. That is why the formula uses a1 + (n-1)d: it counts how far the term is from the start and adds the repeated step size.
When to use it
Use this formula when a list of numbers increases or decreases by the same amount each time, such as in patterns, tables, savings plans, or simple linear sequences. It is useful whenever you need to find a specific term far into the sequence without writing out every term.
Worked example, step by step
Sequence: 12, 17, 22, ... First term: a1 = 12 Common difference: d = 5 Find the 8th term Use a_n = a1 + (n-1)d Substitute the values: a8 = 12 + (8-1)×5 Simplify: a8 = 12 + 7×5 Multiply: a8 = 12 + 35 Add: a8 = 47
Common mistakes to avoid
- Using n instead of n-1, which makes every term too large by one step
- Forgetting to keep the common difference negative when the sequence is decreasing
- Starting from the second term instead of the first term when substituting a1
Did You Know?
Arithmetic sequences model anything that grows by a fixed step — stadium seating, loan payments, stacking bricks.
Share this formula
More in Algebra
View allQuadratic Formula
IntermediateFinds the roots of any quadratic equation ax² + bx + c = 0.
Discriminant
IntermediateDetermines the nature of the roots of a quadratic equation.
Binomial Theorem
IntermediateExpands any power of a binomial into a sum of terms.
Logarithm Product Rule
IntermediateThe logarithm of a product equals the sum of logarithms.
Logarithm Quotient Rule
IntermediateThe logarithm of a quotient equals the difference of logarithms.
Logarithm Power Rule
IntermediateThe logarithm of a power equals the exponent times the logarithm.