PROGRESSION, in mathematics, is either arithmetical or geometrical. Con tinued arithmetic proportion, where the terms do increase and decrease by equal differences, is called arithmetic progres sion.
If 1, 3, 5, 7, 9, &c. a, a + b, a + 2 b, 4 + 3 b, &c. a, a - b, a - '2 b, a - 3 b, &c, are in arithmetical progression. Hence it is manifest, that if a be the first term, and a + b the second, a +2 b is the third, a the fourth, &c. and a + 91 1 .12 the nth or last term: " The sum of a series of quantities in arithmetical progression is found by mul tiplying the sum of the first and last terms by half the number of terms." Let a be the first term, b the common difference, a the number of terms, and 8 the sum of the series : Then, a +a -b +a+2b...a+ 71-1.0=8, or, a+n-l. b+a+a Suna,2a+n-1.6+2a+n--1.6+2a+n-1.6 +&c.
to ton terms, = 2s, or, 2 a+ n - 1.b X =-.-- 23.
as ands=2a+n-1.b 2 Any three of the quantities a, a, n, b, being given, the fourth may be found the equation s = 2 a +2T- 1.6 4 1. To find the sum of 18 terms of the series 1, 3, 5, 7, &e.
Here a = 1, b = 2, a = 18 ; there fore, s = 2 + 34 x 9 = 324.
F.x. 2. Required the sum of 9 terms of the series 11, 9, 7, 5, &c.
In this case a =. 11, b = -2, n = 9 ; 9 therefore s = 22-16 X - =27.
2 Kr. If the first term of an arithmetical progression be 14, and the sum of 8 terms he 28, what is the common differ ence ? Since 2 o+n---1.i 4=8 2 a+ n x 2s n 77=7h = --a = 25-.2 an therefore, b = In the case n. proposed, s .= 28, a = 14, a = 8, there 56-224 - 3.
, = 8 X 7 7 Hence, the series is 14, 11, 8, 5, &c. PROGRESSION geometrical. Quantities are said to be in geometrical progression, or continual proportion, when the first is to the second, as the second to the third, and as the third to the fourth, &c. that is, when every succeeding tern, is a certain multiple, or part of the preceding term. If a be the first term, and ar the second, the series will be a, or, arl," ar3, ar4, &a. For a : or:: ar : ars ar': ar3, &c.
The constant multiplier is called thd common ratio, and it may be found by dividing the second term by the first.
" If quantities be in geometrical pro gression, their differences are in geome trical progression." Let a, ar, ar' ars, arc, &c. be the quan tities; their differences, ar- a, ar. - ar, ar3 - ors - ar4 - ar3, &c. form a geo metrical progression, whose first term is a r - a, and common ratio r.
" Quantities in geometrical progression are proportional to their differences." For a : ar an - a : ors ar :: ar : ar3 - ar', &c.
"In any geometrical progression, the first term is to the third, as the square of the first to the square of the second." Let a, ar, ar', &c. be the progression; then a : or' :: a' : a=r=.
Hence it appears, that the duplicate ratio of two quantities (Enc. Def. 10. 5.), is the ratio of their squares.
In the same manner it may be shown, that the first term is to the n+lth term, as the first raised to the nth power, to the second raised to the same power.
" If any terms be taken at equal in tervals in a geometrical progression, they Will be in geometrical progression." a, a r3n ...... &c. be the progression, then a, a rn, a a &c. are at the interval of a terms, and form a geometrical progression, whose common ratio is ta.
" If the two extremes, and the number of terms in a geometrical progression be given, the means may be found." Let a and b be the eitremes,a the num ber of terms, and r the common ratio; then the progression is a, a r', a r3 a and since b is the last term, a b b = b, and = therefore a a' and r being thus known, the terms of the progression a r, a a r3, &c. are known.
" To find the sum of a series of quan tities in geometrial progression, subtract the first term from the product of the last term and common ratio, and divide the re mainder by the difference between the common ratio and unity." Let a be the first term, r the common ratio, n the number of terms, y the last term, and 8 the sum of the series : Then a±a r+ a r'....+ a a =a ; and multiplying both sides by r, Sub. a-f-ar4-arl-Far3....-1-arn-1 =8 Rem. a + arn= r 8q = r--1 x 8 arna_ rya or, r-1 From the equation 8 any three of the quantities, a, r, y, a, being given, the fourth may be found. When r is a proper fraction, as n increases, the value of rn, or of a rn, decreases, and when n is increased without limit, a rn becomes less, with respect to a, than any magnitude that can be assigned; and a therefore 8= a r r This quantity , which we call the sum of the series, is the limit to which the sum of the terms approaches, but ne ver actually attains ; it is, however, the true representative of the series conti nued sine fine ; for this series arises from the division of a by 1 r ; and therefore a may, without error, be substituted for it.
Ex. 1. To find the sum of 20 terms of the series, 1, 2, 4, 8, &c.
Here a = 1, r = 2, n = 20; therefore, 1 x = 1.
Ex. 2. Required the sum of 12, terms of the series 64, 16, 4, &c.
1 ' Here a = 64, r n = 12, there 64 41 64x4"--64 64 fore, 1 4' 41.-1 4-1 Ex. 3. Required the sum of 12 terms of the series, 1, 3, 9, 27, &c.
In this case, a= 1, r 3, n= 12; 3. 8 = = 3 1 4 Ex. 4. To find the sum of the series 1 2 1 4 1 I 8 in infinition.
1 Here a = 1, r =-- (Art. 224), 1 2 8= 1= 3 1-f-T It may be observed, in connection with this subject, that the recurring decimals are quantities in geometrical progression, 1 1 1 where 1000 &c. is the common ratio, according as one, two, three, &c, figures recur ; and the vulgar fraction, corresponding to such a decimal, is found by summing the series.
Ex. 5. Required the vulgar fraction corresponding to the decima1.123123123, &c.
Let .123123,123, &c._-_---s ; then multi ply both sides by 1000 ; and 123.123123 123, &c. = 10003, and by subtracting the former equation from the latter, 123= 123 41 999s; therefore a =