PROGRESSION, in mathematics, a series or sequence of terms such that each quantity bears the same mathematical relation to the pre ceding. This relation may be either that of a difference or of a ratio; the first case is that of the arithmetical progression, the second that of the geometrical series.
In an arithmetical series each term is de rived from the preceding by adding to it a con stant, called the difference, which may have a plus sign or a minus sign, so that the series may either increase or decrease. In the accepted notation of the arithmetical progression, a is the first term, n the number of terms, d the difference, and 1 the last term. The value of the last term or of any (the nth) term may be determined from the formula 1 = a + ( n 1) d, which is merely a notational statement of the evident fact that the coefficient of d in any term will be less by one than the number of t so that the nth term must be the sum of oft term and of the difference taken (n 1) times Letting s stand for the sum of n terms we may write in ascending or descending order: s = a + (a + d) + (a + 2d) + + (1 d) + 1, or .s=l+ (1d) + (1 2d) + . + (a + d) a.
Adding these two equations we get the equa tion, 2s= (a + 1) + (a + 1) + (a + 1) + + (a + 1) + (a+ 1), or 2s (a +1), it be evident that the factor (a + 1) occurs is times in the second term of the preceding equation. The formula for the sum will then be s.- (a + 1). Hence, if three of data are given as to an arithmetical series, that is, three of a, d, n, 1, s, the fourth (and the fifth, in general) can be found by the regular processes of algebra. Moreover the arithmetical mean be tween two numbers, being the mid-term in a series in which the number of terms is odd, will be one-half the sum of the two numbers.
A geometrical progression is a series in which each successive term is derived from the preceding by multiplying it by a constant, called the ratio, which may be less or greater than unity, so that the series may decrease or in crease, or may be either plus or minus, so that the terms may be either similar or constantly changing in sign. In the accepted notation of
the geometrical progression, a is the first term, r the common ratio, 1 the last (or nth) term and s the sum of the series of Is terms. The formula for 1, the nth term, since the exponent of r increases by one in each term will be 1 P.= To derive the formula for the sum of n terms, get the difference of the two self-evident equations, s = a + ar + ars + + arn I, and rs = ar + ars ? + arn + arn.
The difference is rs s = arn a, or (r 1)s=a ria The formula then is s = or r -1 f - 1 With any three of the quantities given in these two formulas the other two may be derived by the use of quadratic equations (and in one in stance of logarithms). An interesting variation of the problem of the summation of a geometric series is to find the value of a repeating decimal or repetend; as, for example, to find the value of .545454 .. (or .54). Here 0-S4, r°..01, and n = a. As is increases without limit, the value of 1 approaches zero; hence in the formula for rla, the sum, , rl becomes negligible, and the a a formula takes the form or . Substitut r 1 1 r .54 6 ing the values above we have .1.01 .11 A harmonica: progression is a series in which the reciprocals of the terms form an arithmetical series. Any problem relating to harmonical progressions may therefore be treated as problems in arithmetical progression by the mere inversion of each term. See