Prop. 6.

Given the least term, the number of terms, and the common excess, or difference, to find the greatest

term.

Rule. Multiply the number of terms by the common excess, or difference, and to that product add the least term; from this sum subtract the common excess, or difference, and the remainder will be the greatest term.

Note. The following propositions and theorems contain the whole practice of arithmetical progression, (including the propositions and rules already given:)

Proposition 1. Given l, g, and n, to find s and d.

Theo. XVIII. 2g+d+√ 2g+a}2 −8ds__n.

The application of the preceding theorems is very evident and easy. Rules night here be inserted, were they of any use, for finding the sum of polygonal and figurate numbers, constituting part of the ancient Pythagorean speculations about numbers, &c. Should ay person wish to become acquainted with such numbers, he may consult Mr. Malcolm's Arithmetic, from page 396 to 441.

Examples to Proposition 1.

(1.) If the least term of a series of numbers in arithmetical progression be 4, the greatest 100, and the number of terms 17, what is the sum of the terms?

4+100=104, and 104 × 17884, answer.

(2.) If the least term be 3, the greatest 108, and the number of terms 14, what is the sum of the terms?

(3.) How many strokes does the hammer of a clock strike in 12 hours?

(4.) If 100 stones be laid in a straight line, and exactly the space of a yard be left between one stone and another, how far must a person travel who gathers up these stones singly, returning with every one to a basket a yard distant from the first?

Examples to Prop. 2.

(5.) If the least term of a series of numbers in arithmetical progression be 4, the greatest 100, and the number of terms 17, what is the common difference between each term?

100-4-96 divisor, and 17-1-16 dividend, hence 96 divided by 16 gives 6, the common difference.

(6.) If the least term be 3, the greatest 108, and the number of terms 14, what is the common difference?

(7.) A person travelled from London to a certain place in 8 days; he travelled 2 leagues the first day, and every day he travelled farther than he did the preceding by an equal number of leagues; the last day he travelled 23 leagues: how far did he travel every day?

Examples to Prop. 3.

(8.) The least term of a series of numbers in arithmetical progression is 4, the greatest 100, and the common difference between each term is 6; what is the number of terms?

100-4-96 dividend, which divided by 6, gives 16 for the quotient; this increased by an unit, gives 17 for the number of terms.

(9.) If the least term be 3, the greatest 108, and the common difference 5, what is the number of terms?

(10.) A man, going a journey, travelled the first day. 2 leagues, and the last day 23; he increased his journey every day 3 leagues; how many days did he travel?

Examples to Prop. 4.

(11.) The greatest term of a series of numbers in arithmetical progression is 100, the number of terms 17, and the common difference between each term 6; what is the least term?

17—1×6=96; then 100—96—4, answer.

(12.) If the greatest term be 108, the number of terms 22, and the common difference 5, what is the least term?

(13.) A man in 6 days went from London to a certain place; every day's journey was greater than the preceding one by 4 miles; his last day's journey was 40 miles: what was his first?

Examples to Prop. 5.

(14.) The number of terms is 17, the common difference 6, and the sum of the terms, of a series of numbers, in arithmetical progression, is 884; what is the least. term?

884÷÷17–52, and 17—1×6-96; then 52-4, the least term.

(15.) If the number of terms be 22, the common difference 5, and the sum of the terms 1221, what is the least term?

(16.) A man is to receive 3007. at 12 payments, each succeeding payment to exceed the former by 47. What will his first payment be?

Examples to Prop. 6.

(17.) If the least term of a series of numbers in arithmetical progression be 4, the number of terms 17, and the common difference 6, what is the greatest term?

17×6=102, and 102+4=106; then 106—6—100, the greatest

term.

(18.) If the least term be 3, the number of terms 22, and the common difference 5, what is the greatest term? (19.) A man bought 100 yards of cloth; the first yard cost him 28., and each succeeding yard Is. more to the ast; what did the last yard stand him in?

GEOMETRICAL PROGRESSION.

Definition. When a series of numbers increase by a common multiplier, or decrease by a common divisor, those numbers are said to be in geometrical progression ; such as 2, 4, 8, 16, &c. ; or 27, 9, 3, 1, &c. The first and last terms are usually called the extremes, and the common multiplier or divisor the ratio.

Note 1. If three numbers be in geometrical progression, the product of the two extremes will be equal to the square of the mean. Thus, if 3. 9. 27. be in geometrical progression.

Then will 3x27=9×9.

2. If four numbers be in geometrical progression, the product of the two extremes will be equal to the product of the means. Thus, if . 4. 8. 16. be in geometrical progression,

Then will 2x16=4x8.

3. If a series of numbers (consisting of any number of terms) be in geometrical progression, the product of the two extremes will be equal to the product of any two means equidistant from the extremes; or to the square of the mean, if the terms be odd.

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Thus, if 1. 2. 4. 8. 16. 32. &c. be in geometrical progression,
Then will 1x32=2x16=4×8.

Or, if 1. 2. 4. 8. 16. &c. be in geometrical progression,

Then will 1x 16=2×8=4×4.

4. If, out of any series of numbers in geometrical progression, there be taken any series of equidistant terms, that series will likewise be in geometrical progression.

Thus, if 2. 4. 8. 16. 32. 64. &c. be in geometrical progression,
Then will 4. 16. 64. &c. be in geometrical progression.

Proposition 1. Given the number of terms the ratio, and either of the extreme terms, of a limited geometrical series, to find the other extreme.

Rule. Write down a few terms of a geometrical series, beginning with, and formed by, the given ratio; over which place the arithmetical series 1. 2. 3. 4. 5. &c. as indices: observe what figures of these indices, when added together, will give a number an unit less than that expressing the number of terms; and find the product of the terms in the geometrical series which stand under these indices. This product multiplied by the first term given in the question, or the first term divided by this product, according as the progression is increasing or decreasing, will give the term sought.

Prop. 2. Given one extreme, the ratio, and the number of terms, of a geometrical series, to find the sum of the

terms.

Rule. Find the other extreme by Proposition 1. Then divide the difference between the extremes by the ratio less 1; the quotient increased by the greater extreme will give the sum of the terms.