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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_7 for the arithmetic sequence with a_1 = 3 and d = 3.

  1. Worked Example: First identify exactly what the question is asking: Find a_7 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_7 = 3 + (7 - 1)3.
  5. The term is 21.
  6. Check the result by substituting or estimating: the response should match 21 and make sense in the original problem.

Answer: 21

Practice problems

1. Find a_7 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_7 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_7 = 3 + (7 - 1)3.
  5. The term is 21.
  6. Check the result by substituting or estimating: the response should match 21 and make sense in the original problem.

Answer: 21

2. Find a_8 for the arithmetic sequence with a_1 = 4 and d = 4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find a_8 for the arithmetic sequence with a_1 = 4 and d = 4.
  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_8 = 4 + (8 - 1)4.
  5. The term is 32.
  6. Check the result by substituting or estimating: the response should match 32 and make sense in the original problem.

Answer: 32

3. Find a_9 for the arithmetic sequence with a_1 = 5 and d = 5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find a_9 for the arithmetic sequence with a_1 = 5 and d = 5.
  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_9 = 5 + (9 - 1)5.
  5. The term is 45.
  6. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

4. Find a_10 for the arithmetic sequence with a_1 = 6 and d = 6.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find a_10 for the arithmetic sequence with a_1 = 6 and d = 6.
  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_10 = 6 + (10 - 1)6.
  5. The term is 60.
  6. Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.

Answer: 60

5. Find a_6 for the arithmetic sequence with a_1 = 7 and d = 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find a_6 for the arithmetic sequence with a_1 = 7 and d = 2.
  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_6 = 7 + (6 - 1)2.
  5. The term is 17.
  6. Check the result by substituting or estimating: the response should match 17 and make sense in the original problem.

Answer: 17

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