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Finite Series

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 finite series adds a fixed number of terms from a sequence. 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 the finite arithmetic series sum with 6 terms, first term 3, and last term 18.

  1. Worked Example: First identify exactly what the question is asking: Find the finite arithmetic series sum with 6 terms, first term 3, and last term 18.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use S_n = n(a_1 + a_n)/2.
  4. S = 6(3 + 18)/2.
  5. The sum is 63.
  6. Check the result by substituting or estimating: the response should match 63 and make sense in the original problem.

Answer: 63

Practice problems

1. Find the finite arithmetic series sum with 6 terms, first term 3, and last term 18.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the finite arithmetic series sum with 6 terms, first term 3, and last term 18.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use S_n = n(a_1 + a_n)/2.
  4. S = 6(3 + 18)/2.
  5. The sum is 63.
  6. Check the result by substituting or estimating: the response should match 63 and make sense in the original problem.

Answer: 63

2. Find the finite arithmetic series sum with 7 terms, first term 4, and last term 22.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the finite arithmetic series sum with 7 terms, first term 4, and last term 22.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use S_n = n(a_1 + a_n)/2.
  4. S = 7(4 + 22)/2.
  5. The sum is 91.
  6. Check the result by substituting or estimating: the response should match 91 and make sense in the original problem.

Answer: 91

3. Find the finite arithmetic series sum with 8 terms, first term 5, and last term 26.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the finite arithmetic series sum with 8 terms, first term 5, and last term 26.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use S_n = n(a_1 + a_n)/2.
  4. S = 8(5 + 26)/2.
  5. The sum is 124.
  6. Check the result by substituting or estimating: the response should match 124 and make sense in the original problem.

Answer: 124

4. Find the sum of the finite geometric series 3 + 6 + 12 + 24.

Show solution
  1. This is a finite geometric series with first term 3 and ratio r = 2.
  2. Use S_n = a_1(r^n - 1)/(r - 1) = 3(2^4 - 1)/(2 - 1).
  3. 3 * 15 = 45.

Answer: 45

5. Find the sum of the finite geometric series 3 + 6 + 12 + 24. (variation 2)

Show solution
  1. This is a finite geometric series with first term 3 and ratio r = 2.
  2. Use S_n = a_1(r^n - 1)/(r - 1) = 3(2^4 - 1)/(2 - 1).
  3. 3 * 15 = 45.

Answer: 45

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