17. Shew that the coefficient of x in the expansion of enx - 1 1 is {1' + 2 + 3′ + +n'}. Hence find the sum of n ...... 1 e terms of the series 1a + 2a +32 +..., and also of 13 + 23 + 33 + ... 18. Shew that, if a, be the coefficient of x" in the expansion of ee, then 1 1 1 20. Shew that the sum of n terms of the series + + + ...9 1 2 3 beginning at the (n+1)th, becomes equal to log, 2 when n is increased without limit. 22. Prove the following (i) (x + y) − x2 - y2 = 7xy (x + y) (x2 + xy + y2)2, (ii) 11 (x+y)11 — x11 — y11 = 11xy (x + y) (x2 + xy + y2) 13 {(x2 + xy + y2)3 + x2y3 (x + y)3}, {(x2 + xy + y2)3 + 2x3y3 (x + y)2}. (iii) (x + y)13 — x13 — y13 = 13xy (x + y) (x2 + xy + y2)* + ・an-3rf2r+ n (n - r -1)...(n − 3r – 1) where a = x2 + xy + y2 and b = xy (x+y). + 24. Shew that, (i) if n be any uneven integer, (b−c)" (ca)" + (a - b)" will be divisible by (b-c)3 + (c-a)3 + (a - b)3; (ii) if n be of the form 6m 1, it will be also divisible by (b − c)2 + (c − a)2 + (a - b)2; and (iii) if n be of the form 6m+1 it will be divisible by (b−c) + (ca) + (a - b)*. COMMON LOGARITHMS. 307. In what follows the logarithms must always be supposed to be common logarithms, and the base, 10, need not be written. If two numbers have the same figures, and therefore differ only in the position of the decimal point, the one must be the product of the other and some integral power of 10, and hence from Art. 303, II. the logarithms of the numbers will differ by an integer. Thus log 421.5=log 4.215 + log 100 = 2 + log 4.215. Again, knowing that log 3 = 30103, we have log '03 = log (3÷100) = log 3 — log 100 = ·30103 — 2. = Thus On account of the above property, common logarithms are always written with the decimal part positive. log 03 is not written in the form - 169897 but 2-30103, the minus sign referring only to the integral portion of the logarithm and being written above the figure to which it refers. Definition. When a logarithm is so written that its decimal part is positive, the decimal part of the logarithm is called the mantissa and the integral part the characteristic. 308. The characteristic of the logarithm of any number can be written down by inspection. For, if the number be greater than 1, and n be the number of figures in its integral part, the number is clearly less than 10" but not less than 10"-1. Hence its logarithm is between n and n-1: the logarithm is therefore equal to n − 1 + a decimal. Thus the characteristic of the logarithm of any number greater than unity is one less than the number of figures in its integral part. Next, let the number be less than unity. Express the number as a decimal, and let ʼn be the number of ciphers before its first significant figure. Then the number is greater than 10-"-1 and less than 10". Hence, as the decimal part of the logarithm must be positive, the logarithm of the number will be - (n + 1) + a decimal fraction, the characteristic being - (n + 1). Thus, if a number less than unity be expressed as a decimal, the characteristic of its logarithm is negative and one more than the number of ciphers before the first significant figure. For example, the characteristic of the logarithm of 3571-4 is 3, and that of 00035714 is 4. Conversely, if we know the characteristic of the logarithm of any number whose digits form a certain sequence of figures we know at once where to place the decimal point. For example, knowing that the logarithm of a number whose digits form the sequence 35714 is 3.55283, we know that the number must be 3571.4. 309. Tables are published which give the logarithms of all numbers from 1 to 99999 calculated to seven places of decimals: these are called 'seven-figure' logarithms. For many purposes it is however sufficient to use fivefigure logarithms. In all Tables of logarithms the mantissae only are given, for the characteristics can always, as we have seen, be written down by inspection. In making use of Tables of logarithms we have, I. to find the logarithm of a given number, and II. to find the number which has a given logarithm. I. To find the logarithm of a given number. If the number have no more than five significant figures, its logarithm will be given in the tables. But, if the number have more significant figures than are given in the tables, use must be made of the principle that when the difference of two numbers is small compared with either of them, the difference of the numbers is approximately proportional to the difference of their logarithms. This follows at once from Art. 304, for An example will shew how the above principle, called the Principle of Proportional Differences, is utilised. Ex. To find the logarithm of 357.247. We find from the tables that log 3.57245529601, and log 3.5725 ='5529722; and the difference of these logarithms is .0000121. Now the difference between 3.57247 and 3.5724 is ths. of the difference between 3.5724 and 3.5725; and hence if we add ths. of 0000121 to the logarithm of 3.5724 we shall obtain the approximate logarithm of 3.57247. Now ths. of 0000121 is 00000847, which is nearer to 0000085 than to 0000084. Hence the nearest approximation we can find to the logarithm of 3.57247 is ⚫5529601+0000085 ='5529686. The characteristic of the logarithm of 357-247 is obviously 2, and therefore the logarithm required is 2·5529686. II. To find the number which has a given logarithm. For example, let the given logarithm be 4.5529652. We find from the tables that log 3.5724 = 5529601 and that log 3.5725=5529722, the mantissa of the given logarithm falling between these two. Now the difference between 5529601 and the 51 given logarithm is of the difference between the logarithms of 121 3.5724 and 3.5725; and hence, by the principle of proportional 51 differences, the number whose logarithm is 5529652 is 3.5724+ 121 x .0001=3.5724+00004 3.57244. [The approximation could only be relied upon for one figure.] Thus •5529652 = log 3.57244, and therefore 4.5529652 log 000357244. COMPOUND INTEREST AND ANNUITIES. = 310. The approximate calculation of Compound Interest for a long period, and also of the value of an annuity, can be readily made by means of logarithms. All problems of this kind depend upon the three following:-[The student is supposed to be acquainted with the arithmetical treatment of these subjects.] I. To find the amount of a given sum at compound interest, in a given number of years and at a given rate per cent. per annum. Let P denote the principal, n the number of years, 100r the rate per cent. per annum, and A the required amount. Then the interest of P for one year will be Pr, and therefore the amount of principal and interest at the end of the first year will be P (1 + r). This last sum is the capital on which interest is to be paid for the second year; and therefore the amount at the end of the second year will be {P (1+r)}(1+r) = P(1+r)2. Similarly the amount at the end of n years will be P (1+r)". Thus AP (1+r)"; and hence log A = log P+n log (1+r). 2n If the interest is paid, and capitalised half yearly, it can be easily seen that the amount will be P (1+ S. A. 25 |