Ray's New Higher Arithmetic · Arts. 394–396 · Unit 27: Series
118. Geometrical progression
Goal: Your student finds any term and the sum of a geometrical series, including an endless one, and links it to compound interest and annuities.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
- A calculator, for checking only (optional)
What's in the book
Geometrical progression: definition and five quantities, Case I (find an extreme), Case II (find the sum) with two rules, notes connecting compound interest and annuities, examples, finding the ratio and inserting means, and the topical outline of Series. Most answers are printed.
The lesson, step by step
Warm-up
Ask: What comes next: 1, 3, 9, 27, ...? What is happening each time?
The idea
Read:
A Geometrical Progression is a series in which any term after the first is the product of the preceding term by a fixed number.
Thus, 1, 3, 9, 27, is an ascending progression
Case I
Find the 8th term of the series 1, 2, 4, 8, etc.
1 × 2⁷ = 128.
Consider the given extreme as the first term, and multiply it by that power of the ratio whose degree is denoted by the number of terms less one.
Case II
The first term is 3, the ratio 4, the number of terms 5; required the sum of the series.
S = 3 + 12 + 48 + 192 + 768. Then 4S = 12 + 48 + 192 + 768 + 3072. Subtract: 3S = 3072 − 3, so S = 1023.
Find the last term and multiply it by the ratio; then find the difference between this product and the first term, and divide by the difference between the ratio and unity.
Endless series
Try:
Of 1, ½, ¼, etc.
The terms shrink toward 0, so the sum comes as close as you like to 2.
Back to money
Read the book's note: the amount of a debt at compound interest is the last term of a geometrical progression whose ratio is 1 + the rate. Work:
Find the amount of an annuity of $50, the time being 53 yr., the rate per cent 10.
Independent practice and wrap-up
Your student works a few more, including Example 12:
Calculate a table of amounts of an annuity of $1, for any number of years from 1 to 6, at 8%.
Ask: Why does the sum of 1 + ½ + ¼ + ... never pass 2?
Answers to the book's problems
Ex. 12 (no printed answer): amounts of $1 a year at 8%, for 1 to 6 years: $1; $2.08; $3.2464; $4.506112; $5.866601; $7.335929 (each year's amount × 1.08, plus $1). Ex. 11 asks for a proof, not a number.
Worked out for this site from the scan. If a number in your copy differs, trust the book.