PROGRESSIONS

DEFINITIONS.

754. A Progression is a series of numbers so related, that each number in the series may be found in the same manner, from the number immediately preceding it.

755. An Arithmetical Progression is a series of numbers, which increases or decreases in such a manner that the difference between any two consecutive numbers is constant. Thus, 3, 7, 11, 15, 19, 23.

756. A Geometrical Progression is a series of numbers, which increases or decreases in such a manner that the ratio between any two consecutive numbers is constant. Thus, 5, 10, 20, 40, 80, is a geometrical progression.

75%. The Terms of a progression are the numbers of which it consists. The First and Last Terms are called the Extremes and the intervening terms the Means.

758. The Common or Constant Difference of an arithmetical progression is the difference between any two consecutive terms.

759. The Common or Constant Ratio or Multiplier of a geometrical progression is the quotient obtained by dividing any term by the preceding one.

760. An Ascending or Increasing Progression is one in which each term is greater than the preceding one. 761. A Descending or Decreasing Progression is one in which each term is less than the preceding one.

ARITHMETICAL PROGRESSION.

762. There are five quantities considered in Arithmetical Progression, which, for convenience in expressing rules, we denote by letters, thus:

1. A represents the First Term of a progression.

2. L represents the Last Term.

3. D represents the Constant or Common Difference. 4. N represents the Number of Terms.

5. S represents the Sum of all the Terms.

763. Any three of these quantities being given, the other two may be found. This may be shown thus:

Taking 7 as the first term of an increasing series, and 5 the constant difference, the series may be written in two forms; thus:

Observe, in (2), each term is composed of the first term 7 plus as many times the constant difference 5 as the number of the term less 1. Thus, for example, the ninth term in this series would be 7+5 × (9—1)=47. Hence, from the manner in which each term is composed, we have the following formulæ or rules:

A

4. N=4+1.

D

Read, {
{

The number of terms is equal to the last term minus the first term, divided by the common difference, plus 1.

Observe, that in a decreasing series, the first term is the largest and the last term the smallest in the series. Hence, to make the above formulæ apply to a decreasing series, we must place L where A is, and A where L is, and read the formulæ acccordingly.

764. To show how to find the sum of a series let

(3.) 23 +23 + 23 + 23+23+23=twice the sum of the terms.

Now, observe, that in (3), which is equal to twice the sum of the series, each term is equal to the first term plus the last term; hence,

Sof (A + L) x N. Read,

The sum of the terms of an arithmetical series is equal to one-half of the sum of the first and last term, multiplied by the number of terms.

EXAMPLES

FOR

PRACTICE.

765. 1. The first term of an arithmetical progression is 4, the common difference 2; what is the 12th term ?

2. The first round of an upright ladder is 12 inches from the ground, and the nineteenth 246 inches; how far apart are the rounds?

3. The tenth term of an arithmetical progression is 190, the common difference 20; what is the first term?

4. Weston traveled 14 miles the first day, increasing 4 miles each day; how far did he travel the 15th day, and how many miles did he travel in all the first 12 days?

5. The amount of $360 for 7 years at simple interest was $486; what was the yearly interest?

6. The first term of an arithmetical series of 100 terms is 150, and the last term 1338; what is the common difference?

7. What is the sum of the first 1000 numbers in their natural order?

8. A merchant bought 16 pieces of cloth, giving 10 cents for the first and $12.10 for the last, the several prices form an arithmetical series; find the cost of the cloth?

9. A man set out on a journey, going 6 miles the first day, increasing the distance 4 miles each day. The last day he went 50 miles; how long and how far did he travel?

10. How many less strokes are made daily by a clock which strikes the hours from 1 to 12, than by one which strikes from 1 to 24.

GEOMETRICAL PROGRESSION.

766. There are five quantities considered in geometrical progression, which we denote by letters in the same manner as in arithmetical progression; thus:

767. Any three of these quantities being given, the other two may be found. This may be shown thus:

Taking 3 as the first term and 2 as the constant ratio or multiplier, the series may be written in three forms; thus:

Observe, in (3), each term is composed of the first term, 3, multiplied by the constant multiplier 2, raised to the power indicated by the number of the term less 1. Thus, for example, the seventh term would be 3 x 27-13× 26 = 192.

Hence, from the manner in which each term is composed, we have

Ꮮ

4. R-1 =

Α

Read,

The constant multiplier is equal to the root, whose index is indicated by the number of terms less one, of the quotient of the last term divided by the first.

The number of terms less one is equal to the exponent of the power to which the common multiplier must be raised to be equal to the quotient of the last term divided by the first.

768. To show how to find the sum of a geometrical series, we take a series whose common multiplier is known; thus:

S = 5 + 15 + 45 + 135 + 405.

Multiplying each term in this series by 3, the common multiplier, we will have 3 times the sum.

(1.) Sx3=5x3+ 15×3+45 x3+135 x 3+405 × 3, or (2.) Sx3=15 +45 +135 +405 +405 × 3.

Subtracting the sum of the series from this result as expressed in (2), we have,

Sx3 = 15+45 +135+405+405 × 3

Now, observe, in this remainder S x 2 is S× (R − 1), and 405 × 3 is LR, and 5 is A. Hence, S × (R-1) = L × R− A. And since R-1 times the Sum is equal to L × R-A, we have,