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Arithmetic Sequences

A free Precalculus lesson from the “Sequences, Series, and Discrete Models” unit, with a worked example and practice problems including step-by-step solutions.

Arithmetic sequences add the same amount each step. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Sequences and series connect repeated patterns, finance, computer loops, and discrete approximations.

Worked example

Problem. Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.

  1. Worked Example: First identify exactly what the question is asking: Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 + (n - 1)d.
  4. a_5 = 3 + (5 - 1)(3) = 3 + 12.
  5. So a_5 = 15.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

Practice problems

1. Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 + (n - 1)d.
  4. a_5 = 3 + (5 - 1)(3) = 3 + 12.
  5. So a_5 = 15.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

2. Two consecutive terms of an arithmetic sequence are 8 then 12. By how much does the sequence change each step? Give d.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Two consecutive terms of an arithmetic sequence are 8 then 12. By how much does the sequence change each step? Give d.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The common difference is the gap between consecutive terms: d = (later term) - (earlier term).
  4. d = 12 - 8.
  5. So d = 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. The arithmetic sequence 5, 10, 15, 20, ... has common difference 5. Find the next term.

Show solution
  1. Core Practice: First identify exactly what the question is asking: The arithmetic sequence 5, 10, 15, 20, ... has common difference 5. Find the next term.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Add the common difference 5 to the last term shown.
  4. 20 + 5 = 25.
  5. So the next term is 25.
  6. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.

Answer: 25

4. Given the arithmetic sequence 6, 8, 10, ..., find a_8.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Given the arithmetic sequence 6, 8, 10, ..., find a_8.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The first term is a_1 = 6 and the common difference is d = 8 - 6 = 2.
  4. a_8 = 6 + (8 - 1)(2) = 6 + 14.
  5. So a_8 = 20.
  6. Check the result by substituting or estimating: the response should match 20 and make sense in the original problem.

Answer: 20

5. An arithmetic sequence has a_4 = 11 and common difference d = 3. Find the first term a_1.

Show solution
  1. Core Practice: First identify exactly what the question is asking: An arithmetic sequence has a_4 = 11 and common difference d = 3. Find the first term a_1.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Rearrange a_n = a_1 + (n - 1)d to a_1 = a_n - (n - 1)d.
  4. a_1 = 11 - (4 - 1)(3) = 11 - 9.
  5. So a_1 = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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