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Arithmetic Sequences

A free Algebra I lesson from the “Exponential Functions and Sequences” unit, with a worked example and practice problems including step-by-step solutions.

An arithmetic sequence adds the same number d (the common difference) to get from one term to the next. The explicit formula for the n-th term is a_n = a_1 + (n - 1) * d, where a_1 is the first term.

What you'll learn

Why it matters: Saving the same amount each month, simple-interest growth, and any 'add a fixed amount' pattern is arithmetic.

Worked example

Problem. For the sequence 3, 7, 11, 15, ..., find the common difference and the 10th term.

  1. Common difference: 7 - 3 = 4.
  2. Apply the formula: a_10 = 3 + (10 - 1) * 4 = 3 + 36 = 39.

Answer: d = 4, a_10 = 39

Practice problems

1. For 2, 5, 8, 11, ..., find d.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 2, 5, 8, 11, ..., find d.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 5 - 2 = 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. For 2, 5, 8, 11, ..., find a_10.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 2, 5, 8, 11, ..., find a_10.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_10 = 2 + 9 * 3 = 29.
  4. Check the result by substituting or estimating: the response should match 29 and make sense in the original problem.

Answer: 29

3. For 100, 95, 90, ..., find d.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 100, 95, 90, ..., find d.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 95 - 100 = -5.
  4. Check the result by substituting or estimating: the response should match -5 and make sense in the original problem.

Answer: -5

4. For 100, 95, 90, ..., find a_5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For 100, 95, 90, ..., find a_5.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_5 = 100 + 4 * (-5) = 80.
  4. Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.

Answer: 80

5. For 4, 9, 14, 19, ..., find a_6.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For 4, 9, 14, 19, ..., find a_6.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_6 = 4 + 5 * 5 = 29.
  4. Check the result by substituting or estimating: the response should match 29 and make sense in the original problem.

Answer: 29

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