117. Arithmetical progression

The lesson, step by step

1. Warm-up

   Ask: What comes next: 1, 4, 7, 10, ...? What is being added each time?

2. The idea

   Read:

   A Series is any number of quantities having a fixed order, and related to each other in value according to a fixed law.

   An Arithmetical Progression is a series in which any term differs from the preceding or following by a fixed number.

   Thus, 1, 3, 5, 7, 9, is an ascending series, whose common difference is 2;

3. Case I

   Find the 20th term of the arithmetical series 1, 4, 7, 10, etc.

   The difference 3 is added 19 times: 1 + 19 × 3 = 58. Formula: l = a + (n − 1)d. Try:

   Find the 12th term of the series 3, 7, 11, etc.

4. Case II

   What is the sum of 9 terms of the series 1, 4, 7, 10, etc.?

   The book writes the series forward and backward. Each pair adds to 26, and there are 9 pairs, so twice the sum is 234 and the sum is 117.

   Multiply the sum of the extremes by the number of terms, and divide by 2.

5. Guided practice

   Find the sum of the whole numbers 1 to 10000, as one of the book's examples asks. Talk through why the trick works.

6. Exercises in analysis

   Read:

   The following are presented as exercises in analysis

   Try together:

   Find the common difference of a series whose extremes are 8 and 28, and number of terms, 6.

   Insert one arithmetical mean between 8 and 54.

7. Independent practice and wrap-up

   Your student works more of the exercises and checks the printed answers. Ask: How many terms are there from 10 to 500 counting by 5s?