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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 is defined by a_1 = 3 and a_n = a_(n-1) + 3. Find a_3.

  1. This is arithmetic recursion: each term is the one before it plus 3.
  2. a_2 = a_1 + 3 = 3 + 3 = 6; a_3 = a_2 + 3 = 6 + 3 = 9.
  3. So a_3 = 9.

Answer: 9

Practice problems

1. A sequence is defined by a_1 = 3 and a_n = a_(n-1) + 3. Find a_3.

Show solution
  1. This is arithmetic recursion: each term is the one before it plus 3.
  2. a_2 = a_1 + 3 = 3 + 3 = 6; a_3 = a_2 + 3 = 6 + 3 = 9.
  3. So a_3 = 9.

Answer: 9

2. The first term of a sequence is 4, and every later term is found by adding 4 to the term before it. What is the 4th term?

Show solution
  1. Adding 4 repeatedly is the recursion a_n = a_(n-1) + 4.
  2. Term 1 = 4, term 2 = 8, term 3 = 12, term 4 = 12 + 4 = 16.
  3. So the 4th term is 16.

Answer: 16

3. Given a_1 = 5 and the recursive rule a_n = a_(n-1) + 5, list forward to a_5 and report a_5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Given a_1 = 5 and the recursive rule a_n = a_(n-1) + 5, list forward to a_5 and report a_5.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Build one term at a time by adding 5.
  4. a_2 = 10, a_3 = 15, a_4 = 20, a_5 = a_4 + 5 = 20 + 5 = 25.
  5. So a_5 = 25.
  6. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.

Answer: 25

4. A sequence is defined by a_1 = 6 and a_n = 2*a_(n-1). Find a_3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A sequence is defined by a_1 = 6 and a_n = 2*a_(n-1). Find a_3.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. This is geometric recursion: multiply the previous term by 2.
  4. a_2 = 2(6) = 12; a_3 = 2(12) = 24.
  5. So a_3 = 24.
  6. Check the result by substituting or estimating: the response should match 24 and make sense in the original problem.

Answer: 24

5. Each term of a sequence is 3 times the previous term, and the sequence starts at 2. What is the 4th term?

Show solution
  1. Multiplying by 3 each step is the recursion a_n = 3*a_(n-1).
  2. Term 1 = 2, term 2 = 6, term 3 = 18, term 4 = 3(18) = 54.
  3. So the 4th term is 54.

Answer: 54

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