Subtract (1) from (2), and there remains sr-sarn -α. The other terms destroy one another. Hence (r-1)s=a(r”—1). This is the formula for s, when r is greater than unity; but if r is less than unity, the first term of the series will be the greatest, and the proper expression for s is obtained by subtracting (2) from (1), which gives s-sr-a-arn Hence (1-7)s=a(1—"), .. 8= a(1—pn) (4.) If now we represent the last term by l, it is (1) evident that l-arm-1. From these two equations, namely, l=am-1, and s= a(-1) a(1), the following theorems may be de or= 1-r The above theorems are given as exercises in literal analysis, and should all be deduced from the 6th and 9th, which were previously proved. When is less than `1, the term may be rendered less than any quantity that can be named, however small, by taking n sufficiently great; so that (4) in the case of n-infinity, will become 8 1. a which is the expression for the sum of a decreasing geometrical series continued to infinity. EXERCISES. 1. Given the first term of a geometrical series 1, the common ratio 2, and the number of terms 10, to find the last term and the sum of the series. Substitute in Theorems 9th and 6th, and we have Ans. 512, 8=1023. 2. The sum of a geometrical progression, whose first term is 1, and last term 128, is 255. What is the common ratio? Ans. Theorem 4th gives r=2. 3. Find the sum of the geometrical series, 1, 4, 4, 1, &c., continued to infinity. Ans. 2. 4. Find the sum of the geometrical series, whose first term is 1, and common ratio 3, when continued to infinity. Ans. 5. 5. Find the sum of the geometrical series whose first term is m and common ratio when continued to infinity. Ans. mn. n-1 6. A servant agreed with his master to serve him for twelve months, upon this condition, that for his first month's service he should receive a farthing, for the second a penny, for the third fourpence, and so on. What did his wages amount to at the expiration of his service? Ans. L.5825, 8s. 51d. 7. There are three numbers in geometrical progression whose sum is 52, and the sum of the first and second is to the sum of the first and third as 2 is to 5. Required the numbers. Ans. 4, 12, 36. 8. There are three numbers in geometrical progression. The sum of the first and second is 15, and the sum of the first and last is 25. What are the numbers? GEOMETRICAL RATIO. Ans. 5, 10, 20. 97. The geometrical ratio between two numbers is determined by dividing the one number by the other. The quotient is the value of the ratio. The number divided is called the antecedent, and the divisor the consequent of the ratio. Thus the ratio of 9 to 6 is =14, in which 9 is the antecedent and 6 the consequent, and the value of the ratio is 1. Ratio may therefore be considered as a fraction, the numerator of which is the antecedent, and the denominator the consequent of the ratio. When the antecedent is greater than the consequent, it is called a ratio of greater inequality, and when the antecedent is less than the consequent, it is called a ratio of lesser inequality. 98. Since ratios can be expressed by fractions, they can be compared with each other by reducing the fractions to a common denominator; then that will be the greater ratio which has the greater numerator. Ratios are commonly written by placing two points between the antecedent and consequent; thus a:b expresses the ratio of a to b, and is read a is to b. 99. Proposition 1st. A ratio of greater inequality is diminished by adding the same quantity to both its terms; whereas a ratio of lesser inequality is increased by adding the same quantity to each of its terms. a+x For is a ratio of greater inequality, and if c be added a to each of its terms, it becomes a+x+c Reducing these ratios to a common denominator, the first becomes a2+ax+ac a-x a2+ax+ac+cx a(a+c) which is evidently less than the Again, let — be a ratio of lesser inequality; add c to a each of its terms, and it becomes to a common denominator, they become a2-ax+ac a(a+c) by сх a(a+c) and where the second is evidently greater than the first Q. E. D. 100. Proposition 2d. A ratio of greater inequality is increased, and a ratio of lesser inequality diminished, by subtracting the same quantity from each of its terms. a+x Let be a ratio of greater inequality, take c from each be a ratio of lesser inequality; take c from each of its terms, and it becomes ; reducing -ax―ac+cx a(a-c) the former Q. E. D. 01. Prop. 3d. A ratio is not altered by multiplying lividing its terms by the same quantity. et ab be any given ratio, then it is identical with a na nb || Q. E. D. PROPORTION. 02. The equality of two ratios constitutes a proportion; s if a b be equal to c: d, the two constitute a propor1, and are written thus; a:b::c:d, or a:b=c:d, 1 read, a is to b as c is to d; consequently, since the io of a to b is and the ratio of c to d is, a C we have =, in which a and c are called antecedents, and b and d sequents: also a and d are called extremes, and c and neans. Art. 14. 103. Prop. 1. In every proportion the product of the tremes is equal to the product of the means. с For if a: bc:d, then, multiplying both sides bd we have ad-bc. Q. E. D. NOTE. If a:b=b: c, then b is called a mean proportional beeen a and c, and c is called a third proportional to a and b; and y Prop. 1) it is evident that b2=ac; hence b√ac, or a mean oportional between two quantities, is the square root of their oduct. Prop. 2. Two equal products can be converted into a roportion by making the factors of the one product the xtremes, and the other the means. a с For if ad be, dividing both sides by bd, we have ence a: b:: c: d. Q. E. D. 104. Prop. 3. If four quantities be proportional, they re also proportional when taken inversely; that is, the econd is to the first, as the fourth to the third. Since с b d = 1. a 1÷=1 d α с and hence b: a=d: c. Q. E. D. 105. Prop. 4. If four quantities be proportional they are also proportional when taken alternately; that is, the irst is to the third as the second is to the fourth. For b since, if both sides be multiplied by and the com с mon factors cancelled from the numerator and denomina 106. Prop. 5. When four quantities are proportional, they are also proportional by composition; that is, the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth. Since+1+1; hence d Therefore a+b:b=c+d: d. a+b c+d = b d Q. E. D. 107. Prop. 6. When four quantities are proportional, they are also proportional by division; that is, the difference of the first and second is to the second as the difference of the third and fourth is to the fourth. Therefore a—b:b-c-d: d. Q. E. D. 108. Prop. 7. When four quantities are proportional, they are also proportional by mixing; that is, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. a+b c+d For Prop. 5th = and Prop. 6th = b d' ab cad b and therefore a+b: a—b=c+d: cd. Q. E. D. 109. Prop. 8. Quantities are proportional to their equimultiples. Let a and b represent any quantities and ma and mb any equimultiples of them, then a: b=ma: mb. αι ma For therefore a b=ma: mb. Q. E. D. Where m is any quantity, whole or fractional. 110. Prop. 9. The like powers and roots of proportional quantities are proportional. Where n may be either whole or fractional, and consequently represent either a power or a root. 111. Prop. 10. If two proportions have the same antecedents, another proportion may be formed, having the consequents of the one for its antecedents, and the consequents of the other for its consequents. For if a b:: c: d, and a: e::c:f, then, and by inversion,=; |