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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 sum of the arithmetic series with 5 terms, first term a_1 = 3, and last term a_5 = 15.

  1. Worked Example: First identify exactly what the question is asking: Find the sum of the arithmetic series with 5 terms, first term a_1 = 3, and last term a_5 = 15.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. This is an arithmetic series, so use S_n = n(a_1 + a_n)/2.
  4. S = 5(3 + 15)/2 = 5(18)/2.
  5. So the sum is 45.
  6. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

Practice problems

1. Find the sum of the arithmetic series with 5 terms, first term a_1 = 3, and last term a_5 = 15.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the sum of the arithmetic series with 5 terms, first term a_1 = 3, and last term a_5 = 15.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. This is an arithmetic series, so use S_n = n(a_1 + a_n)/2.
  4. S = 5(3 + 15)/2 = 5(18)/2.
  5. So the sum is 45.
  6. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

2. An arithmetic sequence has a_1 = 4 and d = 4. Find the sum of its first 6 terms.

Show solution
  1. First find a_6 = a_1 + (n - 1)d = 4 + (6 - 1)(4) = 24.
  2. This is arithmetic, so S_n = n(a_1 + a_n)/2 = 6(4 + 24)/2.
  3. So the sum is 84.

Answer: 84

3. A finite geometric series has first term a_1 = 1, ratio r = 3, and 3 terms. Find its sum.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A finite geometric series has first term a_1 = 1, ratio r = 3, and 3 terms. Find its sum.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. This is a geometric series, so use S_n = a_1(r^n - 1)/(r - 1).
  4. S = 1(3^3 - 1)/(3 - 1) = 1(27 - 1)/2.
  5. So the sum is 13.
  6. Check the result by substituting or estimating: the response should match 13 and make sense in the original problem.

Answer: 13

4. Find the sum of the first 8 positive integers: 1 + 2 + ... + 8.

Show solution
  1. This is the arithmetic series 1 + 2 + ... + 8 with a_1 = 1 and a_n = 8.
  2. Use S_n = n(a_1 + a_n)/2 = 8(1 + 8)/2.
  3. So the sum is 36.

Answer: 36

5. How many terms are in the series from a_1 = 2 up to a_4 = 11 (an arithmetic series with d = 3)?

Show solution
  1. Core Practice: First identify exactly what the question is asking: How many terms are in the series from a_1 = 2 up to a_4 = 11 (an arithmetic series with 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, so 11 = 2 + (n - 1)(3).
  4. Then (n - 1)(3) = 9, so n - 1 = 3.
  5. So n = 4: there are 4 terms.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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