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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_7 for the arithmetic sequence with a_1 = 3 and d = 3.

  1. Worked Example: First identify exactly what the question is asking: Find a_7 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_7 = 3 + (7 - 1)3.
  5. The term is 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. Unit review 1 (Arithmetic Sequences): Find a_7 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_7 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_7 = 3 + (7 - 1)3.
  5. The term is 21.
  6. Check the result by substituting or estimating: the response should match 21 and make sense in the original problem.

Answer: 21

2. Unit review 2 (Geometric Sequences): Find a_6 for the geometric sequence with a_1 = 4 and r = 4.

Show solution
  1. Unit Review: First identify exactly what the question is asking: Find a_6 for the geometric sequence with a_1 = 4 and r = 4.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 r^(n - 1).
  4. a_6 = 4(4^5).
  5. The term is 4096.
  6. Check the result by substituting or estimating: the response should match 4096 and make sense in the original problem.

Answer: 4096

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

Show solution
  1. Unit Review: 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. Unit review 4 (Explicit Formulas): Use a_n = 3 + (n - 1)5 to find a_8.

Show solution
  1. Unit Review: 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. Unit review 5 (Sigma Notation): In sigma notation, the lower and upper numbers tell you:

Choices: which index values to use · the graph color · the only possible answers · the denominator restrictions

Show solution
  1. Unit Review: First identify exactly what the question is asking: In sigma notation, the lower and upper numbers tell you:
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The index starts at the lower number.
  4. It ends at the upper number.
  5. Each index value is substituted into the rule.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: which index values to use

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