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Recursive 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.

A recursive formula tells how to get the next term from earlier 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. A sequence has a_1 = 2 and a_n = a_(n-1) + 3. Find a_4.

  1. Worked Example: First identify exactly what the question is asking: A sequence has a_1 = 2 and a_n = a_(n-1) + 3. Find a_4.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. a_2 = 5.
  4. a_3 = 8.
  5. a_4 = 11.
  6. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

Practice problems

1. A sequence has a_1 = 2 and a_n = a_(n-1) + 3. Find a_4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A sequence has a_1 = 2 and a_n = a_(n-1) + 3. Find a_4.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. a_2 = 5.
  4. a_3 = 8.
  5. a_4 = 11.
  6. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

2. A sequence has a_1 = 3 and a_n = a_(n-1) + 4. Find a_4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A sequence has a_1 = 3 and a_n = a_(n-1) + 4. Find a_4.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. a_2 = 7.
  4. a_3 = 11.
  5. a_4 = 15.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

3. A sequence has a_1 = 4 and a_n = a_(n-1) + 5. Find a_4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A sequence has a_1 = 4 and a_n = a_(n-1) + 5. Find a_4.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. a_2 = 9.
  4. a_3 = 14.
  5. a_4 = 19.
  6. Check the result by substituting or estimating: the response should match 19 and make sense in the original problem.

Answer: 19

4. A recursive formula is different from an explicit formula because it:

Choices: uses previous term values · always uses logarithms · never needs a starting value · gives only graph intercepts

Show solution
  1. Core Practice: First identify exactly what the question is asking: A recursive formula is different from an explicit formula because it:
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Recursive formulas build from earlier terms.
  4. They need a starting value.
  5. Explicit formulas give a term directly from n.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: uses previous term values

5. A recursive formula is different from an explicit formula because it: (variation 2)

Choices: uses previous term values · always uses logarithms · never needs a starting value · gives only graph intercepts

Show solution
  1. Core Practice: First identify exactly what the question is asking: A recursive formula is different from an explicit formula because it:
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Recursive formulas build from earlier terms.
  4. They need a starting value.
  5. Explicit formulas give a term directly from n.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: uses previous term values

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