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Explicit Formulas

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

An explicit formula lets you jump straight to a term without listing all previous terms. 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. Which explicit formula matches arithmetic sequence a_1 = 4, d = 2?

  1. Worked Example: First identify exactly what the question is asking: Which explicit formula matches arithmetic sequence a_1 = 4, d = 2?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Arithmetic explicit formulas use a_1 + (n - 1)d.
  4. Here a_1 = 4 and d = 2.
  5. Substitute those values into the formula.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: a_n = 4 + (n - 1)2

Practice problems

1. Which explicit formula matches arithmetic sequence a_1 = 4, d = 2?

Choices: a_n = 4 + (n - 1)2 · a_n = 4(2)^(n - 1) · a_n = 2 + (n - 1)4 · a_n = n + 6

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which explicit formula matches arithmetic sequence a_1 = 4, d = 2?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Arithmetic explicit formulas use a_1 + (n - 1)d.
  4. Here a_1 = 4 and d = 2.
  5. Substitute those values into the formula.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: a_n = 4 + (n - 1)2

2. Which explicit formula matches arithmetic sequence a_1 = 5, d = 3?

Choices: a_n = 5 + (n - 1)3 · a_n = 5(3)^(n - 1) · a_n = 3 + (n - 1)5 · a_n = n + 8

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which explicit formula matches arithmetic sequence a_1 = 5, d = 3?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Arithmetic explicit formulas use a_1 + (n - 1)d.
  4. Here a_1 = 5 and d = 3.
  5. Substitute those values into the formula.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: a_n = 5 + (n - 1)3

3. Which explicit formula matches arithmetic sequence a_1 = 6, d = 4?

Choices: a_n = 6 + (n - 1)4 · a_n = 6(4)^(n - 1) · a_n = 4 + (n - 1)6 · a_n = n + 10

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which explicit formula matches arithmetic sequence a_1 = 6, d = 4?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Arithmetic explicit formulas use a_1 + (n - 1)d.
  4. Here a_1 = 6 and d = 4.
  5. Substitute those values into the formula.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: a_n = 6 + (n - 1)4

4. Use a_n = 3 + (n - 1)5 to find a_8.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Use a_n = 3 + (n - 1)5 to find a_8.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute n = 8.
  4. a_8 = 3 + 7(5).
  5. The term is 38.
  6. Check the result by substituting or estimating: the response should match 38 and make sense in the original problem.

Answer: 38

5. Use a_n = 4 + (n - 1)1 to find a_8.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Use a_n = 4 + (n - 1)1 to find a_8.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute n = 8.
  4. a_8 = 4 + 7(1).
  5. The term is 11.
  6. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

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