PROGRESSIONS

536. 1. How does each of the numbers 2, 4, 6, 8, 10, 12, compare with the number that follows it?

2. How may each of the numbers 4, 6, 8, etc., be obtained from the one that precedes it?

3. How does each of the numbers 2, 5, 8, 11, 14, 17, compare with the number that follows it? How with the one that precedes it?

4. Write in succession some numbers beginning with 3 having a common difference of 2.

5. Write a series of numbers beginning at 4, and having a common difference of 4.

6. Write a series of numbers beginning with 25, and decreasing by the common difference 4.

7. How does each of the numbers 2, 4, 8, 16, 32, etc., compare with the one that follows it? How may each be obtained from the one that precedes it?

8. Write a series of numbers beginning with 2 and increasing by a common multiplier 3.

9. Write a series of numbers beginning with 5, and increasing by a common multiplier 5.

DEFINITIONS.

537. A Series of numbers is numbers in succession, each derived from the preceding according to some fixed laws.

538. The first and last terms of a series are called the extremes, the intervening terms the means.

Thus, in the series 2, 4, 6, 8, 10, the numbers 2 and 10 are the extremes and the others are the means.

539. An Ascending Series is one in which the numbers increase regularly from the first term.

Thus, 2, 5, 8, 11, 14, 17, 20, etc., is an ascending series.

540. A Descending Series is one in which the numbers decrease regularly from the first term. Thus, 48, 24, 12, 6, 3, is a descending series.

ARITHMETICAL PROGRESSION.

541. An Arithmetical Progression is a series of numbers which increase or decrease by a constant common difference.

Thus, 5, 9, 13, 17, 21, etc., is an arithmetical progression of which the common difference is 4.

542. 1. The first term of an arithmetical series is 3 and the common difference is 2. What is the 7th term?

PROCESS.

Com. diff., 2 X 6 = 12

First term, 3+12=15, the 7th term.

ANALYSIS. Since the common difference is 2, the second term is equal to the first plus once the common differ

ence, the third term is equal to the first plus twice the common difference, the fourth term is equal to the first term plus three times the common difference. Hence, the seventh term will be equal to the first term plus six times the common difference, which is 15.

RULE. Any term of an arithmetical progression is equal to the first term, increased by the common difference multiplied by a number one less than the number of terms.

2. The first term is 10 and the common difference 5. What is the 10th term? Prove it.

3. The first term is 6 and the common difference is 8. What is the 25th term?

4. A boy agreed to work for 50 days at 25 cents the first day, and an increase of 3 cents per day. What were his wages the last day?

5. A body falls 16 feet the first second, 3 times as far the second second, 5 times as far the third second. far will it fall the seventh second?

How

6. An arithmetical series has 1000 terms, the first term of which is 75 and the common difference 5. What is the last term?

7. Find the sum of an arithmetical series of which the first term is 2, the common difference 3, and the number of terms 7.

PROCESS.

2+(6×3)=20 2+20 22 22 2 11 11X 7 77

ANALYSIS. By examining the series 2, 5, 8, 11, 14, 17, 20, it is evident that the average term is 11, for if half the sum of any two terms equidistant from the extremes be found it will be 11, and in general in any arithmetical progression the average term is equal half the sum of the extremes or any two terms equidistant from the extremes. Since the first term is 2 and the common difference 3, the last term is found by the previous rule to be 20. The sum of the extremes is therefore 22, which, divided by 2, gives the average term. And since there are 7 terms, the sum will be 7 times the average term, or 77.

RULE. To find the sum of an arithmetical series: Multiply half the sum of the extremes by the number of terms.

8. What is the sum of an arithmetical series composed of 50 terms, of which the first term is 2 and the common difference 3?

9. What is the sum of a series in which the first term is the common difference, 1, and the number of terms 100?

10. A man walked 15 miles the first day, and increased his rate 3 miles per day for 10 days. How far did he walk in the eleven days?

11. How many strokes does a clock strike in 12 hours?

12. A person had a gift of $100 per year from his birth until he became 21 years old. These sums were deposited in a bank and drew simple interest at 6%. How much was due him when he became of age?

GEOMETRICAL PROGRESSION.

543. A Geometrical Progression is a series of numbers which increase or decrease by a constant multiplier or ratio.

Thus, 5, 10, 20, 40, 80; etc., is a geometrical progression, of which the multiplier or ratio is 2.

WRITTEN EXERCISES.

544. 1. The first term of a geometrical series is 3 and the multiplier or ratio is 2. What is the 5th term?

PROCESS.

24-16 3X16 48

ANALYSIS. Since the multiplier is 2, the second term will be 3X2, the third 3×2×2 or 3X2, the fourth 3X 22 X2 or 3 X 23 and the fifth 323 × 2 or 3 × 21, that is, the fifth term is equal to the first term multiplied by the ratio raised to the fourth power.

RULE.—Any term of a geometrical progression is equal to the first term, multiplied by the ratio raised to a power one less than the number of the term.

2. The first term of a geometrical progression is 10, the ratio 3. What is the 6th term?

3. The first term of a geometrical progression is 10, the ratio 4, and the number of terms 6. What is the 6th term? 4. If a farmer should hire a man for 10 days, giving him 5 cents for the first day, 3 times that sum for the second day, and so on, what would be his wages for the last day?

5. If the first term is $100 and the ratio 1.06, what is the 6th term? Or, what is the amount of $100 at compound interest for 5 years at 6%?

6. What is the amount of $520 for 6 years, at 5% compound interest?

7. What is the sum of a geometrical series, of which the first term is 5, the ratio 3, and the number of terms 5?

RULE.—The sum of a geometrical series is equal to the difference between the first term, and the product of the last term by the ratio, divided by the difference between the ratio and 1.

8. The extremes of a geometrical progression are 4 and What is the sum of the series?

1024, and the ratio 4.

9. The extremes are is the sum of the series?

10. What is the sum of the series in which the first term is 2, the last term 0, and the ratio ; or what is the sum of the infinite series 2, 1, 1, 1, 1, 16, 32, etc?