PROGRESSION

, an orderly advancing or proceeding in the same manner, course, tenor, proportion, &c.

Progression is either Arithmetical, or Geometrical.

Arithmetical Progression, is a series of quan<*>ities proceeding by continued equal differences, either increasing or decreasing. Thus,

increasing	1,	3,	5,	7,	9,	&c,	or
decreasing	21,	18,	15,	12,	9,	&c;

where the former progression increases continually by the common difference 2, and the latter series or Progression decreases continually by the common difference 3.

1. And hence, to construct an arithmetical Progression, from any given first term, and with a given common difference; add the common difference to the first term, to give the 2d; to the 2d, to give the 3d; to the 3d, to give the 4th; and so on; when the series is ascending or increasing: but subtract the common difference continually, when the series is a descending one.

2. The chief property of an arithmetical Progression, and which arises immediately from the nature of its construction, is this; that the sum of its extremes, or first and last terms, is equal to the sum of every pair of intermediate terms that are equidistant from the extremes, or to the double of the middle term when there is an uneven number of the terms.

Thus,	1,	3,	5,	7,	9,	11,	13,
	13,	11,	9,	7,	5,	3,	1,
Sums	14	14	14	14	14	14	14

where the sum of every pair of terms is the same number 14.

Also,	a,	a + d,	a + 2d,	a + 3d,	a + 4d,
	a + 4d,	a + 3d,	a + 2d,	a + d,	a
sums	2a + 4d	2a + 4d	2a + 4d	2a + 4d	2a + 4d

3. And hence it follows, that double the sum of all the terms in the series, is equal to the sum of the two extremes multiplied by the number of the terms; and consequently, that the single sum of all the terms of the series, is equal to half the said product. So the sum of the 7 terms| 1, 3, 5, 7, 9, 11, 13, is . And the sum of the five terms , is .

4. Hence also, if the first term of the Progression be 0, the sum of the series will be equal to half the product of the last term multiplied by the number of terms: i. e. the sum of , where n is the number of terms, supposing 0 to be one of them. That is, in other words, the sum of an arithmetical Progression, whether finite or insinite, whose first term is 0, is to the sum of as many times the greatest term, in the ratio of 1 to 2.

5. In like manner, the sum of the squares of the terms of such a series, beginning at 0, is to the sum of as many terms each equal to the greatest, in the ratio of 1 to 3. And

6. The sum of the cubes of the terms of such a series, is to the sum of as many times the greatest term, in the ratio of 1 to 4.

7. And universally, if every term of such a Progression be raised to the m power, then the sum of all those powers will be to the sum of as many terms equal to the greatest, in the ratio of m + 1 to 1. That is, the sum , is to , in the ratio of 1 to .

8. A synopsis of all the theorems, or relations, in an arithmetical Progression, between the extremes or first and last term, the sum of the series, the number of terms, and the common difference, is as sollows: viz, if a denote the least term, z the greatest term, d the common difference, n the number of terms, s the sum of the series; then will each of these five quantities be expressed in terms of the others, as below: . And most of these expressions will become much simpler if the first term be 0 instead of a.

Geometrical Progression, is a series of quantities proceeding in the same continual ratio or proportion, either increasing or decreasing; or it is a series of quantities that are continually proportional; or which increase by one common multiplier, or decreafe by one common divisor; which common multiplier or divisor is called the common ratio. As,

increasing,	1,	2,	4,	8,	16,	&c,
decreasing,	81,	27,	9,	3,	1,	&c;

where the former progression increases continually by the common multiplier 2, and the latter decreases by the common divisor 3. Or ascending, a, ra, r²a, r³a, &c, or descending, a, a/r, a/r², a/r³, &c; where the first term is a, and common ratio r.

1. Hence, the same principal properties obtain in a geometrical Progression, as have been remarked of the arithmetical one, using only multiplication in the geometricals for addition in the arithmeticals, and division in the former for subtraction in the latter. So that, to construct a geometrical Progression, from any given first term, and with a given common ratio; multiply the 1st term continually by the common ratio, for the rest of the terms when the series is an ascending one; or divide continually by the common ratio, when it is a descending Progression.

2. In every geometrical Progression, the product of the extreme terms, is equal to the product of every pair of the intermediate terms that are equidistant from the extremes, and also equal to the square of the middle term when there is a middle one, or an uneven number of the terms.

Thus,	1,	2,	4,	8,	16,
	16	8	4	2	1
prod.	16	16	16	16	16

Also	a,	ra,	r²a,	r³a,	r⁴a,
	r⁴a	r³a	r²a	ra	a
prod.	r⁴a²	r⁴a²	r⁴a²	r⁴a²	r⁴a²

3. The last term of a geometrical Progression, is equal to the first term multiplied, or divided, by the ratio raised to the power whose exponent is less by 1 than the number of terms in the series; so z = ar^{n - 1} when the series is an ascending one, or , when it is a descending Progression.

4. As the sum of all the antecedents, or all the terms except the least, is to the sum of all the consequents, or all the terms except the greatest, so is 1 to r the ratio. For,| if be all except the last, then are all except the first; where it is evident that the former is to the latter as 1 to r, or the former multiplied by r gives the latter. So that, z denoting the last term, a the first term, and r the ratio, also s the sum of all the terms; then , or . And from this equation all the relations among the four quantities a, z, r, s, are easily derived; such as, ; viz, multiply the greatest term by the ratio, subtract the least term from the product, then the remainder divided by 1 less than the ratio, will give the sum of the series. And if the least term a be 0, which happens when the descending Progression is infinitely continued, then the sum is barely . As in the infinite Progression &c, where z = 1, and r = 2, it is s or .

5. The first or least term of a geometrical Progression, is to the sum of all the terms, as the ratio minus 1, to the n power of the ratio minus 1; that is .

Other relations among the five quantities a, z, r, n, s, where a denotes the least term, z the greatest term, r the common ratio, n the number of terms, s the sum of the Progression, are as below; viz, . And the other values of a, z, and r are to be found from these equations, viz, .

For other sorts of Progressions, see Series.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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