Ray's New Elementary Algebra · Arts. 226–230 · Unit 8: Chapter VIII: Progressions and proportion

52. Geometrical progression

Goal: Your student finds any term and the sum of a geometrical series, the sum of an infinite decreasing series, and a mean proportional.

⏱ About 45 minutes · 8 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to the page
  • Pencil and plenty of paper (or a whiteboard)
  • A card to cover printed answers

What's in the book

Geometrical progression defined; the rules for any term, the sum, and the sum of an infinite decreasing series; finding the ratio from the first and last terms; and exercises with printed answers.

The lesson, step by step

  1. Warm-up

    Ask: What comes next: 1, 2, 4, 8, ...? 54, 18, 6, ...?

  2. Definition

    Read Art. 226.

    A Geometrical Progression is a series of terms, each of which is derived from the preceding, by multiplying it by a constant quantity, termed the ratio.

  3. Any term

    Read Art. 227.

    Multiply the first term by the ratio raised to a power, whose exponent is one less than the number of terms.

    Example 1: 4 × 3⁴ = 324. Your student does:

    2. Find the 6th term of the progression 2, 8, 32, etc. Ans. 2048.

  4. The sum

    Read Art. 228 and the clever subtraction that proves it.

    Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder by the ratio less one.

    Your student does exercise 2: 7 terms of 1, 2, 4, ... (127).

  5. Infinite series

    Read Art. 229.

    Divide the first term by one minus the ratio.

    Work together:

    7. A body moves 10 ft. the first sec, 5 the next, 2 1/2 the next, and so on forever; how many ft. would it move over? Ans. 20.

  6. Mean proportional

    Read Art. 230. Between 4 and 16 the middle term is 8. Your student finds a mean proportional between 8 and 32 (16).

  7. Independent practice

    Art. 227 exercises 3 to 7, Art. 228 exercises 3 and 6, Art. 229 exercises 2 and 3, and Art. 230 exercises 1 and 4. The site's calculator (/homeschool/calculator/) can check large powers like 4 × 3⁹ after the setup is done by hand.

  8. Wrap-up

    Ask: Why does 10 + 5 + 2½ + ... never pass 20?

Tip

Fold a sheet of paper in half again and again to show a decreasing geometric series: half, a quarter, an eighth, and the pieces never exceed the whole sheet.

Answers to the book's problems

The book prints the answer after each exercise.

Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.

@teddyrookbook GitHub