Ray's New Practical Arithmetic · Arts. 264–267 · Unit 26: Progressions
121. Arithmetical progression
Goal: Your child finds the last term, the common difference and the sum of an arithmetical series.
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 331 defines Arithmetical Progression (Art. 264) and begins Case I (Art. 265). Page 332 finishes Case I and starts Case II (Art. 266). Page 333 gives Case III (Art. 267), the sum, and page 334 its last problems. Answers are printed.
The lesson, step by step
Warm-up
Count by 4s starting at 3: 3, 7, 11, 15 ... What is the 10th number? Can you find it without counting?
The idea
An Arithmetical Progression is a series of numbers which increase or decrease by a common difference.
The numbers forming the series are called terms; the first and last terms are the extremes; the other terms, the means.
Last term
I bought 10 yd. of muslin, at 3 ct. for the first yard, 7 ct. for the second, 11 ct. for the third, and so on: what did the last yard cost?
The 10th yard adds 4 ct. nine times: 3 + 36 = 39 ct.
1. Multiply the common difference by the number of terms less one. 2. If an increasing series, add the product to the first term; if a decreasing series, subtract the product from the first term.
Difference and sum
Divide the difference of the extremes by the number of terms less one.
For the sum, write the series forward and backward (1, 3, 5, 7, 9, 11 over 11, 9, 7, 5, 3, 1). Each pair makes 12, and there are 6 pairs: 72, which is twice the sum. So the sum is 36.
Multiply the sum of the extremes by the number of terms; and take half the product.
Guided practice
Do these together:
Find the last term of an increasing series in which the first term is 2, the common difference 3, and the number of terms 50.
How many strokes does the hammer of a clock strike in 12 hours?
Independent practice and check
Your child works the remaining problems on pages 332–334, including the 100 apples and the falling body. 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.