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Sigma Notation

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

Sigma notation is compact adding language: lower bound, upper bound, and rule for each term. 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. Evaluate the sum from k = 1 to 5 of k.

  1. Worked Example: First identify exactly what the question is asking: Evaluate the sum from k = 1 to 5 of k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Expand the sum as 1 + 2 + ... + the upper limit.
  4. 1 + 2 + ... + 5 = 15.
  5. That is the finite sum.
  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. Evaluate the sum from k = 1 to 5 of k.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate the sum from k = 1 to 5 of k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Expand the sum as 1 + 2 + ... + the upper limit.
  4. 1 + 2 + ... + 5 = 15.
  5. That is the finite sum.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

2. Evaluate the sum from k = 1 to 6 of k.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate the sum from k = 1 to 6 of k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Expand the sum as 1 + 2 + ... + the upper limit.
  4. 1 + 2 + ... + 6 = 21.
  5. That is the finite sum.
  6. Check the result by substituting or estimating: the response should match 21 and make sense in the original problem.

Answer: 21

3. Evaluate the sum from k = 1 to 7 of k.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluate the sum from k = 1 to 7 of k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Expand the sum as 1 + 2 + ... + the upper limit.
  4. 1 + 2 + ... + 7 = 28.
  5. That is the finite sum.
  6. Check the result by substituting or estimating: the response should match 28 and make sense in the original problem.

Answer: 28

4. 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. Core Practice: 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

5. In sigma notation, the lower and upper numbers tell you: (variation 2)

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

Show solution
  1. Core Practice: 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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