Ray's New Elementary Algebra · Arts. 220–225 · Unit 8: Chapter VIII: Progressions and proportion

51. Arithmetical progression

Goal: Your student finds any term and the sum of an arithmetical series, and inserts means between two numbers.

⏱ 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

Series and arithmetical progression defined; the rule for any term, the rule for the sum (shown by writing the series forward and backward), the general problem, inserting means, and many exercises. Answers are printed.

The lesson, step by step

  1. Warm-up

    Ask: What comes next: 1, 3, 5, 7, ...? 30, 27, 24, ...? How do you know?

  2. Definitions

    Read Arts. 220 and 221.

    An Arithmetical Progression is a series of quantities which increase or decrease, by a common difference.

  3. Any term

    Read Art. 222. The nth term is l = a + (n − 1)d.

    1. For an Increasing Series.—Multiply the common difference by the number of terms less one, and add the product to the first term.

    Example 1: first term 3, difference 5: the 8th term is 3 + 7 × 5 = 38.

  4. The sum

    Read Art. 223: write 3, 5, 7, 9, 11 forward and backward; each pair adds to 14; five pairs make 70, so the sum is 35.

    Multiply half the sum of the two extremes by the number of terms.

  5. Guided practice

    7. a=30, and d=— 2 ; required the 8th term. Ans. 16.

    1. Find the sum of the natural series of numbers 1, 2, 3, 4, .... carried to 1000 terms. Ans. 500500.

  6. Inserting means

    Read Art. 225: five means between 3 and 15. With 7 terms, d = (15 − 3) ÷ 6 = 2: 3, 5, 7, 9, 11, 13, 15.

  7. Independent practice

    Art. 222 exercises 3 to 9, Art. 223 exercises 2 to 5, and Art. 225 exercises 2, 3, 5 and 8. (Exercise 3: a clock strikes 1092 times in a week.)

  8. Wrap-up

    Ask your student to find the sum of the first 100 odd numbers without adding them. (10000.)

Tip

Writing the series forward and backward, as Ray does, is worth doing with real numbers first. Once seen, the sum rule is never forgotten.

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