(d.) The ratio of each term to that which follows it is called the COMMON RATIO, and is always the number by which we multiplied to produce the series. (e.) If it be an increasing series, the common ratio will equal a whole number or an improper fraction; but if it be a decreasing series, the common ratio will equal a proper fraction. (f) From the method of forming such series, it is obvious that the second term must equal the first multiplied by the common ratio; that the third term must equal the first multiplied by the second power of the common ratio; &c. (g.) Hence, any term of a geometrical series must equal the product of the first term multiplied by the common ratio raised to a power one degree less than the number of the term. (h.) Moreover, if the last term of a geometrical series be divided by the first, the quotient will be the common difference raised to a power one degree less than the number of the term. 237. Problems. 1. What is the 7th term of the series of which 125 is the 1st term, and 2 the common ratio? 2. What is the 5th term of the series of which 1 is the 1st term and the common ratio? 3. What is the 9th term of the series of which 4096 is the 1st term and the common ratio? 4. What is the common ratio of the series of which 1 is the 1st and 81 the 5th term? 5. Construct the series of which 1 is the 1st and 6561 is the 7th term. 6. Construct a series of 8 terms, having 1 for the 1st term and for the common ratio. 238. To find the Sum of a Geometrical Series. (a.) If each term of a geometrical series should be multiplied by the common ratio, a new series would be formed, of which the first term would equal the second term of the former series, the second term would equal the third of the former, &c.; the last term but one of the new series would equal the last of the given series. Hence, the first term of the original series would have no corresponding term in the derived series, and the last term of the derived series would have no correspond. ing term in the original series. Thus, by multiplying each term of the series 2, 6, 18, &c., to 1458 by the common ratio, we have - 2 6 18 54 162 486 1458 = given series. 6 18 54 162 486 1458 4374 derived series = 3 times given series. 1 Again. Multiplying each term of the series 32, 8, 2, by 128 (b.) Now, as the given series equals once itself, and the derived series equals the common ratio times the given series, it follows that the difference between the given and derived series will equal the product of the given series multiplied by the difference between 1 and the common ratio. (c.) But, as has been shown, the difference between the series equals the difference between the first term of the given series and the last term of the derived series. (d.) Hence, to find the sum of a geometrical series, we may multiply the last term by the common ratio, and divide the difference between the product and the first term by the difference between 1 and the common ratio. 239. Problems. 1. What is the sum of the series of which 2 is the 1st term, 1458 the last, and 3 the common ratio? Solution. 3 times 1458 = 4374, from which subtracting 2 leaves 4372. Dividing this by 3 1, or 2, gives 2186 for the sum of the series. 2. What is the sum of the series of which 32 is the 1st term, the last, and the common ratio? Solution. TX = 1024, which taken from 32 leaves 311821. This equals 1 , or, times the required sum. Hence, the sum of the series equals 31182÷4241}. = 3. What is the sum of the series of which is the 1st term, 486 the last, and 3 the common ratio? 4. What is the sum of a series of 11 terms of which 16 is the 1st term and the common ratio? 5. What is the sum of a series of 5 terms of which 5 is the 1st term and 3125 the last term? 240. Infinite Decreasing Series. (a.) As in a decreasing series each term is smaller than the preceding, it follows that if the series be carried far enough, the terms will become so small that they may be disregarded without affecting sensibly the sum of the series. (b.) An infinite decreasing series will always be of this character, and hence its sum will equal the quotient obtained by dividing the first term by the difference between 1 and the common ratio. 1. What is the sum of the infinite decreasing series of which 4 is the 1st term and the common ratio? What is the sum of the infinite decreasing series 2. Of which 1 is the 1st term and 3. Of which 3 is the 1st term and 4. Of which 2 is the 1st term and 5. Of which .37 is the 1st term ratio? the common ratio? the common ratio? the common ratio? and .01 the common 6. Of which .597 is the 1st term and .001 the common ratio? 7. Of which .2794 is the 1st term and .0001 the common ratio? SECTION XIX. 241. CIRCULATING DECIMALS. (a.) A circulating or repeating decimal is one which will never terminate, but in which the same figure, or succession of figures, will always follow each other in the same order. Examples. .9999, &c.; .323232, &c.; .0200602006, &c. .5174351743, &c. ; (b.) Circulating decimals are equivalent to vulgar frac tions, the exact decimal value of which cannot be found. See 143, (a.) (c.) That such vulgar fractions must give rise to repeating decimals may be shown thus: Since the remainder after any division must be less than the divisor, we shall at some stage of the division, explained in 143, find a remainder equal to a former remainder, and from this point the quotients and remainders will succeed each other in the same order as before. (d.) A repeating decimal is indicated by placing a dot over the repeating figure, or over the first and last figures of the repeating period. (e.) The repeating part of a decimal will begin as soon as all the factors of 10 have been cancelled from the denominator of the vulgar fraction which produces it, the vulgar fraction being in all cases reduced to its lowest terms. See 143, (b.) 1. With which place will the repetend of the decimal value of begin? Answer. Since 12 contains the third power of 2, which is a factor of 10, the repetend will commence after 3 places have been obtained, i. e., with the fourth place. 2. With which place will the repetend of commence ? Answer. Since 7 contains no factor of 10, the repetend will com mence with the first place. With which place will the repetend of each of the following fractions commence ? (f) An expression which contains or'y the figures of the repetend is, called a SINGLE REPETEND; one which contains other figures is called a MIXED REPETEND. Thus, .427 is a simple repetend, and .53427 is a mixed repetend. (g.) Two repetends are similar when they begin at the same decimal place, and conterminous when they end at the same decimal place. Thus, 523 and .9, or .2473, and .417 are similar; .427 and .436, or .4279, and 0372 are conterminous. (h.) The repeating period may be considered as beginning at any figure, provided that it is made to include the entire combination which is repeated. Thus, 417 = 4174 = .41741 = give .417417417417, &c. .417417; for each developed would (i.) A repeating decimal is really an infinite decreasing series in geometrical progression, of which the first term is the first repeating period, and the common ratio is the decimal fraction, having 1 for a numerator and the power of 10, whose exponent contains as many units as there are places in the repeating period, for its denominator. (See last three problems in 239.) Thus, 48 a series of which .48 is the 1st term and .01 the com mon ratio. Hence, .48 = .48 ÷ (1 — .01) = .48 ÷ .99 = §§. Again. 479: = a series of which .479 is the 1st term and .001 the common ratio. Hence, .479 = .479 (1.001) = = .479.999 = (j) We should reach the same result by observing that = = } = .111, &c. .i; 's ói ; = .00010001, &c. = .0001; &c. §; 8 = 8 x .i=}; 27=27 × .01; 4949 × .oi = 1; X 子;.045= 45 9, &c.; = 4657488; .328923.329933. (k.) Hence, it follows that every repeating decimal is equivalent to a vulgar fraction, of which the numerator is expressed by the repeating period, and the denominator by as many 9's as there are figures in the repeating period. (2.) In multiplying a repeating decimal by any multiple or power of ten, care must be taken to fill the places left vacant by the change of the point by the figures of the repeating period, and to observe what figures would have to be added to any period on account of the multiplication of the preceding period. |