We can obtain the same result by going back to the proposed 7, which, in the particular case progression abc. of q=1, reduces to series is equal to na. The result 0 0 а: а: а: given by the formula, may be regarded as indicating that the series is characterized by some particular property. In fact, the progression, being entirely composed of equal terms, is no more a progression by quotients than it is a progression by differences. Therefore, in seeking for the sum of a certain number of the terms, there is no reason for using the formula (a+1)n S= a(q”—1), in preference to the formula S= 9-1 gives the sum in the progression by differences. 2 which which represents the sum of n of its terms, can be put under the form α aq" Now, since the progression is decreasing, q is a proper fraction; and q′′ is also a fraction, which diminishes as n increases. Therefore the greater the number of terms we take, the more will α ×q" diminish, and consequently the more will the partial Չ sum of these terms approximate to an equality with the first part α of S, that is, to Finally, when n is taken greater than 1-q a any given number, or n= ∞, then -q Xq" will be less than any given number, or will become equal to 0; and the expression a will represent the true value of the sum of all the terms of 1-q the series. Whence we may conclude, that the expression for the sum of the terms of a decreasing progression, in which the number of terms is infinite, is This is, properly speaking, the limit to which the partial sums approach, by taking a greater number of terms in the progression. The difference between these sums and α can become as small as we please, and will only become nothing when the number We have for the expression of the sum of the terms The error committed by taking this expression for the value of the sum of the n first terms, is expressed by the sum of a certain number of terms, is less in proportion as this 196. The consideration of the five quantities a, q, n, 1, and S, which enter into the formulas l-aq"-1 and S= lq-a q-1 (Arts. 191 and 192), give rise to several curious problems. Of these cases, we shall consider here only the most important We will first find the values of S and q in terms of a, l, and n. Substituting this value in the second formula, the value of S will be obtained. The expression q= n-1 the following question, viz.: furnishes the means for resolving To find m mean proportionals between two given numbers a and b ; that is, to find a number m of means, which will form with a and b, considered as extremes, a progression by quotients. For this purpose, it is only necessary to know the ratio. Now, the required number of means being m, the total number of terms is equal to m+2. Moreover, we have l=b, therefore the value m+1/b of becomes q= that is, we must divide one of the given numbers (b) by the other (a), then extract that root of the quotient whose index is one more than the required number of means. Thus, to insert six mean proportionals between the numbers 3 and 384, we make m=6, whence REMARK. When the same number of mean proportionals are inserted between all the terms of a progression by quotients, taken two and two, all the progressions thus formed will constitute a single progression. CHAPTER V. Formation of Powers, and Extraction of Roots of any degree whatever. 197. The resolution of equations of the second degree supposes the process for extracting the square root to be known; in like manner the resolution of equations of the third, fourth, &c. degree, requires that we should know how to extract the third, fourth, &c. root of any numerical or algebraic quantity. It will be the principal object of this chapter to explain the raising of powers, the extraction of roots, and the calculus of radicals. Although any power of a number can be obtained from the rules of multiplication, yet this power is subjected to a certain law of composition which it is absolutely necessary to know, in order to deduce the root from the power. Now, the law of composition of the square of a numerical or algebraic quantity, is deduced from the expression for the square of a binomial (Art. 117); so likewise, the law of a power of any degree, is deduced from the same power of a binomial. We will therefore determine the development of and power of a binomial. 198. By multiplying the binomial x+a into itself several times, the following results are obtained; (x+a)=x+a, (x+a)2=x2+2ax+a2, (x+a)3=x3+3ax2+3a2x+a3, (x+a)=x+5ax1+10a3x3+10a3x2+5a*x+a3 By inspecting these developments it is easy to discover a law according to which the exponents of x and a decrease and increase in the successive terms; it is not, however, so easy to discover a law for the co-efficients. Newton discovered one, by means of which, any power of a binomial can be formed, without first obtain. ing all of the inferior powers. He did not however explain the course of reasoning which led him to the discovery of it; but the existence of this law has since been demonstrated in a rigorous Of all the known demonstrations of it, the most elementary is that which is founded upon the theory of combinations. How. ever, as it is rather complicated, we will, in order to simplify the exposition of it, begin by resolving some problems relative to combi. nations, from which it will be easy to deduce the formula for the binomial, or the development of any power of a binomial. manner. Theory of Permutations and Combinations. 199. Let it be proposed to determine the whole number of ways in which several letters, a, b, c, d, &c. can be written one after the other. The results corresponding to each change in the position of any one of these letters, are called permutations. Thus, the two letters a and b furnish the two permutations ab and ba. |