Ray's New Practical Arithmetic · Arts. 268–270 · Unit 26: Progressions

122. Geometrical progression

Goal: Your child finds the last term and the sum of a geometrical series.

⏱ About 35 minutes · 7 steps

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You'll need

  • Paper and pencil, or a slate or small whiteboard

What's in the book

Page 334 defines Geometrical Progression (Art. 268) and gives Case I (Art. 269). Page 335 works two examples and problems and gives Case II (Art. 270). Page 336, the last page of the book, works the sum and has the final problems. Answers are printed.

The lesson, step by step

  1. Warm-up

    Start with 1 and keep doubling: 1, 2, 4, 8 ... How big is the 10th number?

  2. The idea

    A Geometrical Progression is a series of numbers increasing by a common multiplier, or decreasing by a common divisor.

    The common multiplier or common divisor, is called the ratio.

    Examples from the book: 1, 3, 9, 27, 81 (ratio 3) and 48, 24, 12, 6, 3 (ratio 2).

  3. Last term

    The first term of an increasing geometric series, is 2; the ratio, 3: what is the fifth term?

    2 × 3 × 3 × 3 × 3 = 2 × 3⁴ = 162.

    1. Raise the ratio to a power whose exponent is one less than the number of terms.

    Then multiply the first term by it (or divide, for a decreasing series).

  4. Sum

    Find the sum of 5 terms of the geometric series, whose first term is 4, and ratio 3.

    Write 4 + 12 + 36 + 108 + 324, and below it the series times 3, shifted one place: 12 + 36 + 108 + 324 + 972. The difference, 972 − 4 = 968, is twice the sum, so the sum is 484.

    Multiply the greatest term by the ratio; from the product subtract the least term; divide the remainder by the ratio less 1.

  5. Guided practice

    Do these together:

    The first term is 10; the ratio, 3; the number of terms, 7: what is the sum of the series?

    A father gave his daughter on New Year's day $1; he doubled it the first day of every month for a year: what sum did she receive?

  6. Independent practice and check

    Your child works the remaining problems on pages 335–336. The last few are about infinite decreasing series. Read the remark together first:

    When a series is decreasing, and the number of terms infinite, the last term is 0.

    The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.

  7. Celebrate

    This is the last page of the book. Look back through the chapters together and let your child choose a favorite problem to work again.

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