EZ Math

Precalculus · Sequences and series

How to Find the nth Term of an Arithmetic Sequence

Problem

3,7,11,15,…3, 7, 11, 15, \ldots

Answer

a10=39a_{10} = 39

An arithmetic sequence adds the same number each time. That number is the common difference, and it controls every later term.

Step-by-step solution

  1. Find the common difference.

    d=7−3=4d = 7 - 3 = 4

    Why: Each term adds the same number.

  2. Write the nth term formula.

    an=a1+(n−1)da_n = a_1 + (n - 1)d

    Why: The first term starts the count, and each later term adds one more d.

  3. Substitute the first term, the difference and n = 10.

    a10=3+(10−1)×4a_{10} = 3 + (10 - 1) \times 4

    Why: There are nine steps from the first term to the tenth.

  4. Calculate.

    a10=3+36=39a_{10} = 3 + 36 = 39

    Why: The tenth term is 39.

Common mistakes

  • Using n instead of n - 1. That gives one extra step.
  • Subtracting the terms in the wrong order and getting d = -4.
  • Multiplying the first term by n.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. 2,5,8,…, 10th term2, 5, 8, \ldots,\ \text{10th term}

    Show answer

    2929

  2. 1,4,7,…, 8th term1, 4, 7, \ldots,\ \text{8th term}

    Show answer

    2222

  3. 10,7,4,…, 6th term10, 7, 4, \ldots,\ \text{6th term}

    Show answer

    −5-5

  4. 5,10,15,…, 12th term5, 10, 15, \ldots,\ \text{12th term}

    Show answer

    6060

  5. −2,1,4,…, 7th term-2, 1, 4, \ldots,\ \text{7th term}

    Show answer

    1616