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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. Use the explicit formula a_n = a_1 + (n - 1)d with a_1 = 3 and d = 3 to find a_7.

  1. Worked Example: First identify exactly what the question is asking: Use the explicit formula a_n = a_1 + (n - 1)d with a_1 = 3 and d = 3 to find a_7.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The explicit formula jumps straight to a_7 without listing earlier terms.
  4. a_7 = 3 + (7 - 1)(3) = 3 + 18.
  5. So a_7 = 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. Use the explicit formula a_n = a_1 + (n - 1)d with a_1 = 3 and d = 3 to find a_7.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Use the explicit formula a_n = a_1 + (n - 1)d with a_1 = 3 and d = 3 to find a_7.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The explicit formula jumps straight to a_7 without listing earlier terms.
  4. a_7 = 3 + (7 - 1)(3) = 3 + 18.
  5. So a_7 = 21.
  6. Check the result by substituting or estimating: the response should match 21 and make sense in the original problem.

Answer: 21

2. For the arithmetic sequence with explicit formula a_n = 4 + (n - 1)(4), find a_5.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For the arithmetic sequence with explicit formula a_n = 4 + (n - 1)(4), find a_5.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute n = 5 directly into the explicit formula.
  4. a_5 = 4 + (5 - 1)(4) = 4 + 16.
  5. So a_5 = 20.
  6. Check the result by substituting or estimating: the response should match 20 and make sense in the original problem.

Answer: 20

3. Use the explicit formula a_n = a_1 * r^(n - 1) with a_1 = 2 and r = 3 to find a_3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Use the explicit formula a_n = a_1 * r^(n - 1) with a_1 = 2 and r = 3 to find a_3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. A geometric explicit formula uses a power of r, not a multiple of d.
  4. a_3 = 2 * 3^(3 - 1) = 2 * 9.
  5. So a_3 = 18.
  6. Check the result by substituting or estimating: the response should match 18 and make sense in the original problem.

Answer: 18

4. Which explicit formula matches the arithmetic sequence with a_1 = 6 and d = 2?

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

Show solution
  1. Arithmetic sequences ADD the common difference, so the form is a_1 + (n - 1)d.
  2. Here a_1 = 6 and d = 2.
  3. That gives a_n = 6 + (n - 1)(2).

Answer: a_n = 6 + (n - 1)(2)

5. Which explicit formula matches the geometric sequence with a_1 = 4 and r = 3?

Choices: a_n = 4 * 3^(n - 1) · a_n = 4 + (n - 1)(3) · a_n = 4 + 3^(n - 1) · a_n = 4 * 3^n

Show solution
  1. Geometric sequences MULTIPLY by the common ratio, so the form is a_1 * r^(n - 1).
  2. Here a_1 = 4 and r = 3.
  3. That gives a_n = 4 * 3^(n - 1).

Answer: a_n = 4 * 3^(n - 1)

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