number have The word sig- generally used in its 416. In the solution of the preceding problems we generally used the word number in its largest sense, as nifying fractional as well as whole numbers, and we have con- largest sidered the solution of problems as practicable and possible, sense. when the unknown numbers sought for were comprehended in this extended meaning of the term: many cases however will present themselves in which the term number must be understood in its most limited sense, as signifying whole numbers only, and where the occurrence of fractions or incommensurate numbers would be altogether incompatible with the declared conditions of the problem: thus, if the problem proposed, "to find a number, in the series of natural numbers, the double of whose square should exceed three times the number itself by 5," we should find that the only value of x in the corresponding equation 2 2x2 - 3x = 5, would be which is a fraction and not one of the series of natural numbers, and that consequently the problem is impossible in the precise sense in which it was proposed. Again, if the problem proposed "to find a number consisting of two digits, which, when divided by the sum of its digits, gives a quotient greater by 2 than its first digit; but if its digits were inverted and the resulting number divided by a number greater by unity than the sum of its digits, the quotient is greater by 2 than the one obtained before," we should find, upon representing the two digits in their order by x and y, that the corresponding values of x and y would be 2 and 4, and 14 133 100 133 respectively, the second of which must be rejected, as not being included in the meaning of the word number as limited to the expression of one of the nine digits. obtained in blems are 417. The discrepancies, such as those just noticed, between The results the results of algebraical and other operations and the strict the soluinterpretation which they must sometimes receive in order to tion of proanswer the conditions of a problem, will be confined in Arith- sometimes foreign to metical Algebra, whenever the results are possible and therefore the proobtained, to our enlarged use of the word number: in Symbo- blems pro posed. lical Algebra, however, when operations become practicable, under all circumstances, by the independent use of signs of M M affection, which must be interpreted with a mixed reference to the general conditions which they are required to fulfil and the specific nature of the magnitudes or symbols to which they are prefixed, we shall perpetually meet with results which are foreign to the problem in whose solution they originate, and which are incapable of any interpretation with immediate reference to it: thus we shall find a symbolical result in the solution of those simple equations which we have characterized as impossible, inasmuch as they require the performance of an operation which is not possible in arithmetic, (Art. 380): and in all quadratic equations we shall find two roots or values of the unknown symbol, not only in those cases which we have considered to be unambiguous, but likewise in those which are arithmetically impossible, (Art. 386): but it must be kept in mind, that we are not thus enabled to extend the range of the arithmetical solution, or to obtain any results which will answer the arithmetical conditions of the problem, in a proper arithmetical sense, which are not equally obtainable by the more limited methods which we have pursued in this chapter. CHAPTER VI. ON ARITHMETICAL, GEOMETRICAL AND HARMONICAL PRO- of an arith progres 418. A SERIES of numbers, consisting of any number of Definition terms, which continually increase or diminish by equal dif- metical ferences, is termed an Arithmetical Series or Progression. Thus, series or the series of natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, sion. &c. forms an Arithmetical Progression, since they continually increase by unity and the same series of numbers in an inverted order 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, forms equally an Arithmetical Progression, because they continually diminish by the same number. common and the character of the series be known, 419. The number, by which the successive terms of an If the first Arithmetical Series is increased or diminished, is called their term, the common difference; and it will follow that if the first term, the difference common difference and the character of the series, whether increasing or decreasing, be known, we may form any number of its successive terms. Thus, if the first term of an increasing arithme- its terms tical series be 4 and if the common difference be 3, we readily form formed to may be the series 4, 7, 10, 13, 16, 19, 22, &c.: and if the first term any extent. of a decreasing arithmetical series be 30 and the common difference be 4, the series will be 30, 26, 22, 18, 14, 10, 6, 2: it will not be possible to continue it farther than the last term 2, since we cannot subtract a greater number (4) from a less (2)*. 420. If we assume a to represent the first term, and b the General common difference, of an arithmetical series, the series itself, if increasing, will be represented by representation and formation of arithmetical series. a, a+b, a+2b, a+3b, In Symbolical Algebra, this operation is possible and the Series may be continued indefinitely, the succeeding terms being - 2, - 6, 8, &c.: the same remark applies to all decreasing series. and if decreasing by (1) (2) (3) (4) a, a - b, a-2b, a-3b, a-4b, a - 5b, &c. We have placed the numbers (1), (2), (3), (4), (5), (6), &c. above the successive terms to designate their position with reference to the first, and it will be evident both from inspection and from the least consideration of the successive formation of these terms, that the coefficient of the multiple of the common difference which is added to or subtracted from, the first term, in order to form any assigned term of the series, will be less by unity than the number which denominates the position of the term: thus, in the third term Expression this coefficient is 2, in the sixth term it is 5, and in the nth term, it will be n1: if, therefore, the whole number of terms in the series be denoted by n, the last term of the first series will be represented by a + (n − 1)b, and of the second series by a − (n − 1) b. of the th term. Mode of representing a series of ʼn terms. 421. The usual mode of representing an entire series of n terms is, to write down the first, second and as many more terms at the beginning of the series, as are sufficient to explain the law of its formation, and also its last term, interposing between them a series of dots to indicate the intermediate and deficient terms: thus, an increasing arithmetic series whose first term is a, common difference b, and the number of whose terms is n, would be written thus, or a + (a + b) + (a + 2b) +... .... ... .... {a + (n − 1) b} ; two terms of the beginning of the series being sufficient to indicate the law of its formation, when the character of the series is known, and three of them being generally sufficient in other cases when the nature of the series, whether it is arithmetical or not, is not previously known: thus, the series of n natural numbers, beginning from 1, would be represented by the series of n odd numbers beginning from 1, would be represented by 1+ 3+. (2n-1), In a similar manner, the continued product of n natural numbers beginning from 1 would be represented by These modes of representing a series of terms, which follow a law which is either previously expressed, or which is required to be inferred from the examination of a sufficient number of its terms and no more, are not only extremely convenient, but absolutely necessary, in all general reasonings concerning the summation and properties of such series. sum of an 422. If the several terms of an arithmetical series be added Investigation of a together, we may represent their sum by s, and it will be easy rule for to investigate a simple rule by which such a sum (s) may be finding the determined by a shorter process than the aggregation of all its arithmetic terms, and more particularly so, if the number of those terms be considerable: for it is evident that the sum of an arithmetical series will be the same, when the same series of terms are written in a direct and a reverse order, and therefore we have s= a - + (a + b) + (a+2b) +....{a+(n−1)b}. (1), s = {a + (n − 1) b} + {a + (n − 2) b} + {a+(n−3) b} + ........a......(2); where the first and last terms of the first series, the second and last term but one, the third and last term but two, the fourth and last term but three, and so on, are written severally underneath each other in the two series: consequently, if we add the two series (1) and (2) together, term by term, we shall get 2s = {2a + (n-1)b} + {2a + (n − 1)b} + ... {2a + (n − 1) b}, which is a series of n identical terms, each equal to 2a + (n − 1)b, or to the sum of the first and last terms of the original series (1): it is therefore evident, that the sum of these n equal terms will be equal to n times one of them, or that 2s = {2a + (n − 1) b} n, series. |