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Unit 7 Review and Quiz

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

This checkpoint verifies discrete-model fluency before the final bridge topics. 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. Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.

  1. Worked Example: First identify exactly what the question is asking: Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 + (n - 1)d.
  4. a_5 = 3 + (5 - 1)(3) = 3 + 12.
  5. So a_5 = 15.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

Practice problems

1. Unit review 1 (Arithmetic Sequences): Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.

Show solution
  1. Unit Review: First identify exactly what the question is asking: Find a_5 for the arithmetic sequence with a_1 = 3 and d = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 + (n - 1)d.
  4. a_5 = 3 + (5 - 1)(3) = 3 + 12.
  5. So a_5 = 15.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

2. Unit review 2 (Geometric Sequences): A geometric sequence has a_1 = 4 and r = 4. Find the second term a_2.

Show solution
  1. Unit Review: First identify exactly what the question is asking: A geometric sequence has a_1 = 4 and r = 4. Find the second term a_2.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_2 = a_1 * r.
  4. a_2 = 4 * 4.
  5. So a_2 = 16.
  6. Check the result by substituting or estimating: the response should match 16 and make sense in the original problem.

Answer: 16

3. Unit review 3 (Recursive Formulas): 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. Unit Review: 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. Unit review 4 (Explicit Formulas): 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. Unit review 5 (Sigma Notation): Expand and add: the sum from k = 1 to 8 of k.

Show solution
  1. Unit Review: First identify exactly what the question is asking: Expand and add: the sum from k = 1 to 8 of k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute k = 1, 2, ..., 8 into the rule.
  4. That is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8.
  5. Adding the terms gives 36.
  6. Check the result by substituting or estimating: the response should match 36 and make sense in the original problem.

Answer: 36

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