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Arithmetic and Geometric Series

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

A series is the sum of the terms of a sequence. For arithmetic series with first term a_1, last term a_n, and n terms: S_n = (n/2)(a_1 + a_n). For geometric series with first term a_1 and common ratio r: S_n = a_1 * (1 - r^n) / (1 - r). When |r| < 1 the geometric series has an infinite sum S = a_1 / (1 - r). Sigma notation writes a series compactly: SUM from k = 1 to n of f(k).

What you'll learn

Why it matters: Loan amortization (geometric series), retirement saving with monthly contributions, fractal lengths (Koch snowflake), and zeno-style approximations all use series.

Worked example

Problem. Find the sum of the first 10 terms of 3, 7, 11, 15, ...

  1. It is arithmetic: a_1 = 3, d = 4, so a_10 = 3 + 9 * 4 = 39.
  2. S_10 = (10 / 2)(3 + 39) = 5 * 42.
  3. = 210.

Answer: 210

Practice problems

1. Arithmetic 2, 4, 6, 8, ... Find the sum of the first 10 terms.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Arithmetic 2, 4, 6, 8, ... Find the sum of the first 10 terms.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_10 = 2 + 9*2 = 20.
  4. S_10 = (10/2)(2 + 20) = 5 * 22 = 110.
  5. Check the result by substituting or estimating: the response should match 110 and make sense in the original problem.

Answer: 110

2. Arithmetic 1, 3, 5, 7, ... Find the sum of the first 10 terms.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Arithmetic 1, 3, 5, 7, ... Find the sum of the first 10 terms.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_10 = 1 + 9*2 = 19.
  4. S_10 = (10/2)(1 + 19) = 100.
  5. Check the result by substituting or estimating: the response should match 100 and make sense in the original problem.

Answer: 100

3. Geometric 1, 2, 4, 8, ... Find the sum of the first 5 terms.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Geometric 1, 2, 4, 8, ... Find the sum of the first 5 terms.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. S_5 = 1 * (1 - 2^5)/(1 - 2) = (1 - 32)/(-1) = 31.
  4. Check the result by substituting or estimating: the response should match 31 and make sense in the original problem.

Answer: 31

4. Geometric 2, 6, 18, 54. Find S_4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Geometric 2, 6, 18, 54. Find S_4.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_1 = 2, r = 3.
  4. S_4 = 2 * (1 - 81)/(1 - 3) = 2 * (-80)/(-2) = 80.
  5. Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.

Answer: 80

5. Infinite geometric series 1 + 1/2 + 1/4 + ... Find the sum.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Infinite geometric series 1 + 1/2 + 1/4 + ... Find the sum.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. a_1 = 1, r = 1/2.
  4. S = 1 / (1 - 1/2) = 1 / (1/2) = 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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