Transposing (x+y) in Equation (1), and (x2+y2) in Equation The left hand members of equations (4) and (5) are equal, there 'fore, Or (as)2-2pb-s2+2p a2-2as+2s2-4p=b Clear equation (3) of fractions, and 3+y=ap-ps. (6) (7) Put this value of p in equation (6) and reduce, we have, Taking the given values of a and b we have, 1582+858-70X15 Or 38+178210, an equation which gives s=6. That is, x+y=6, and xy=8, from which we find x=2, and y=4; therefore, the required numbers are 1, 2, 4 and 8, Ans. 3. The arithmetical mean of two numbers exceeds the geo metrical mean by 13, and the geometrical mean exceeds the harmonical mean by 12. What are the numbers? Let x and y represent the numbers. - Then (x+y)= the arithmetical mean, √y the geome trical mean, (Art. 112.) and 2xy x+y the harmonical mean. Let a=12; Then, by the question, (x+y)=√xy+a+1 (1) Take the value of s from equation (3) and put it into equation (4), dividing the numerator and the denominator by 2, and we Clearing of fractions, we shall have p+a√p+√p=p+a√p+(a+1)a Drop equals, and p=(a+1)a (6) Put this value of p in equation (3) and we have Or s=(a+1)a+(a+1)=(a+1)(a-+1) s=2(a+1)2 (7) For the sake of brevity, put (a+1)=b; squaring equation (6) and restoring the values of s and p in equations (6) and 7), And x2-2xy+y2=4b2(b2—a2)=4b2(b+a)(b—a) (C) As a=12 and b=13, b+a=25, and b—a=1. By evolution, x-y=2b×5 Or By subtraction, 2x=262+10b x= b2+ 5b=(b+5)b=18×13=234 2y=262-106 y= b2 5b=(b-5)b- 8x13=104. A more brief solution is the following: Let xy and x+y represent the numbers. the arithmetical mean, x2-y2 the geometri Then The right hand members of equations (1) and (2) being the Put this value of y2 in equation (1), and by squaring x2—26x+(13)2=x2—25x, or x=(13)2=169. Hence, y=65, and the numbers are 104 and 234. 4. Divide the number 210 into three parts, so that the last shall exceed the first by 90, and the parts be in geometrical progression. Ans. 30, 60, and 120. 5. The sum of four numbers in geometrical progression is 30; and the last term divided by the sum of the mean terms is 13. What are the numbers? Ans. 2, 4, 8, and 16. 6. The sum of the first and third of four numbers in geometrical progression is 148, and the sum of the second and fourth is 888. What are the numbers? Ans. 4, 24, 144, and 864. 7. It is required to find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Ans. 2, 4, and 8. s. There are four numbers in geometrical progression, the second of which is less than the fourth by 24; and the sum of the extremes is to the sum of the means, as 7 to 3. What are the numbers ? Ans. 1, 3, 9 and 27. 9. The sum of four numbers in geometrical progression is equal to the common ratio +1, and the first term is are the numbers? What What are the Ans. 12,8, and 6. 10. The sum of three numbers in harmonical proportion is 26, and the product of the first and third is 72. numbers? 11. The continued product of three numbers in geometrical progression is 216, and the sum of the squares of the extremes is 328. What are the numbers? Ans. 2, 6, 18. 12. The sum of three numbers in geometrical progression is 13, and the sum of the extremes being multiplied by the mean, the product is 30. What are the numbers? Ans. 1, 3, and 9. 13. There are three numbers in harmonical proportion, the sum of the first and third is 18, and the product of the three is 576. What are the numbers? Ans. 6, 8, 12. 14. There are three numbers in geometrical progression, the difference of whose difference is 6, and their sum 42. What are the numbers? Ans. 6, 12, 24 15. There are three numbers in harmonical proportion, the difference of whose difference is 2, and three times the product of the first and third is 216. What are the numbers? Ans. 6, 8, and 12. 16. Divide 120 dollars between four persons, in such a way, that their shares may be in arithmetical progression; and if the second and third each receive 12 dollars less, and the fourth 24 dollars more, the shares would then be in geometrical progression. Required each share. Ans. Their shares were 3, 21, 39, and 57, respectively. 17. There are three numbers in geometrical progression, whose sum is 31, and the sum of the first and last is 26. What are the numbers? Ans. 1, 5, and 25. 18. The sum of six numbers in geometrical progression is 189, and the sum of the second and fifth is 54. What are the numbers? Ans. 3, 6, 12, 24, 48 and 96. 19. The sum of six numbers in geometrical progression is 189, and the sum of the two means is 36. What are the numbers? Ans. 3, 6, 12, 24, 48 and 96. CHAPTER III. PROPORTION. (Art. 176.) We have given the definition of geometrical proportion in (Art. 41.) and demonstrated the most essential property, the equality of the products between extremes and means. We now propose to extend our investigations a little farther. Proportion can only exist between magnitudes of the same kind, and the number of times and parts of a time, that one measures another, is called the ratio. Ratio is always a number, and not a quantity. (Theorem 1.) If two magnitudes have the same ratio as two others, the first two as numerator and denominator may form one member of an equation; and the other two magnitudes as numerator and denominator will form the other member. Let A and B represent the first two magnitudes and r their ratio. Also C and D the other two magnitudes, and r their ratio. (Theorem 2.) Magnitudes which are proportional to the same proportionals, are proportional to each other. |