2a+( 2a+(-1)d. Xn. By substituting the values of the letters. 2 Example 7. The first term =10, and the common differ 2a+(n−1)dx n. stituting the values of the letters, s= 2 Sub 2×10+(47-1)×20 2 X47 = 20+920 940 X 47- = 2 X 47=220,90-220 dollars and 90 cents what was paid for the books. Seven problems under Article 433. I. Let x the second of the four numbers, and y=their common difference: Then 1. The series =x—y, x, x+y, x+2y 2. By the problem, x-y+x+x+y+x+2y=56 3. By the second condition, (x-y)2+x2+(x+y)2+(x+ 2y)2=864 4. Uniting terms in the 2d, 4x+2y=56 5. Expanding in the 3d, x2-2xy+y2+x2+x2+2xy+y2+ x2+4xy+4y2=864 6. Uniting terms, 4x2+4xy+6y2=864 7. Dividing by 2 in the 4th, 2x+y=28 8. Transposing 2x, y=28-2x 9. Substituting y's value in the 6th, 4x2+4x(28-2x)+6 (28-2x)2=864 10. Exp. 4x2+112x−8x2+4704-672x+24x2 =864 11. Transposing and uniting, 20x2 — 560x— — -3840 12. Dividing by 20, x2-28x=-192 13. Completing the square, x2-28x+196—192+196 14. Extracting the square root, x-14=±2, and x=12 15. Substituting a's value in the 8th, y=28-24-4. Therefore the 4 numbers are 8, 12, 16, 20. II. Let x=the 2d term, and y-the common difference. 1. The series =x−y, x, x+Y, A 2. By the given conditions, x-y+x+x+y=9 (x-y)3+x3+(x+y)=153 4. Ex.x3+3xy2 -3x2y—y2+x3+x3+3x2y+3xy2+y2=153 5. Uniting terms, 3x3+6xy2-153 6. Uniting terms in the 2d, 3x=9, and x=3 7. Substi. x's value in the 5th, 3×27+6×3×y2=153 8. Multiplying, 81+18y2=153 9. Transposing and uniting, 18y2=72, y2=4, and y=2. Therefore the series is 1, 3, 5. III. Let x=the 2d term, and y=the common difference. 1. x−y, x, x+y=the series. 2. By the proposed conditions, x−y+x+x+y=15 (x-y)2+(x+y)=58 4. Expanding, x2 −2xy+y2+x2+2xy+y2=58 5. Uniting terms, 2x2 +2y2=58 6. Dividing by 2, x2+y2=29 7. Uniting terms and dividing in the 2d, 3x=15, and x=5. 8. Substituting ax's value in the 6th, 25+y2=29 9. Transposing and extracting, y2=4, and y=2. Therefore x=5, and y=2, x—y, x, x+y= 3, 5, 7=the series. IV. Let x= the 2d term, and y= the common difference : then, 1. x−y, x, x+y, x+2y= the series. 2. By the first condition, (x-y)2+x2=34 3. By the second (x+y)2+(x+2y)2=130 4. Expanding in the 2d equation, x2 -2xy+y2+x2=34 5. Expanding in the 3d 4y2=130 66 x2+2xy+y2+x2+4xy+ 6. Uniting terms in the 4th, 2x2-2xy+y2=34 10. Dividing by 8, x2+y2=29 11. Trans. y2, and evolving, x2=29-y2, and x= √29-y2 12. Subst. for x in the 6th, 2×29—y2 — 2y √29—y2+y2=34 13. Expanding, 58-2y2-2y√29-y2+y2=34 14. Transposing and uniting, -2y√29—y2 —y2 —24 15. Involving to the 2d power, 4y2 29-y3y4-48y+576 16. Expanding and uniting, -5y++164y2=576 17. Changing signs, 5y4-164y2 = −576 18. Completing the square 100y4-3280y2+2689611520+26896 19. Extracting the square root, 10y164-124 20. Transposing and uniting, 10y2 =40, y2=4, and y=2 21. Substituting y's value in the 10th, 2-4+29, and x=5. Therefore, x-y, x, x+y, x+2y=3, 5, 7, 9= the series. The above process, from the 7th to the 17th step, may be abridged as follows: 8. Subtracting the 6th from the 7th, 8xy+4y=96 9. Transposing and dividing by 8y, x= 12 y y 2 ·24+2+72-3y2+5y2=130 y2 2 y2 288 11. Transposing and uniting, 2y2+ 82 2 y2 12. Clearing of fractions and uniting, 5y4-164y2 — — 576 The remaining part of the process will be the same as from the 17th step in the solution which has been given. V. Let x-y, x, and x+y= the three digits. 1. By the problem, 100 (x-y)+10x+(x+y)= the local value. the 2. " 100(x-y)+10x+(x+y)÷3x= 26. 3. Multiplying by 3x and uniting, 111x-99y=78x 4. By the problem, 111x-99y+198=100(x+y)+10x+ (x-y) 5. Uniting terms, 111x-99y+198=111x+99y 6. Transposing and uniting, 198y=198, and y=1. 7. Substituting y's value in the 3d, 111x-99-78x, and x=3. Therefore x-y, x, x+y=2, 3, 4, and 234=the number. VI. Let x-3y=the 1st term, and 2y= the common difference. 1. x—3y, x—y, x+y, x+3y= the series. 2. By the 1st condition, (x-3y)+(x+3y)2=200 3. By the 2d (x—y)2 +(x+y)2=136 4. Expanding in the 2d, x2-6xy+9y+x+6xy+9y2=200 5. Expanding in the 3d, x2-2xy+y2+x2+2xy+y2 =136 6. Uniting terms in the 4th, 2x2+18y2=200 7. Uniting terms in the 5th, 2x2+2y2=136 8. Subtract. the 7th from the 6th, 16y2=64, y2=4, and y=2. 9. Substituting y's value in the 7th, 2x2+2×4=136 10. Transposing and uniting, 2x2 =128, and x=8 Therefore x-3y, x−y, x+y, x+3y=2, 6, 10, 14 the series. VII. Let x- -3y=the 1st term, and 2y= the common dif. 1. x-3y, x—y, x+y, x+3y= the series 2. By the problem, 4x=28, and x=7. 3. " 66 66 (x-3y)x(x-y)×(x+y)×(x+3y)=585 4. Substi. 7 for x, (7—3y)× (7—y)×(7+y) × (7+3y)=585 5. Expanding, 2401-490y2+9y4=585 6. Transposing and uniting, 9y4-490y2=—1816 7. Completing the square, 324y4-17640ya +240100=65376+240100 8. Extracting the square root, 18y2-490=+418 9. Transposing and uniting, 18y2 =72, y2 =4, and y=2. Therefore, x-3y, x-y, x+y, x+3y=1, 5, 9, 13 the series. GEOMETRICAL PROGRESSION. Two problems under article 440. 1 2 I. By the formula m+1=r. a = Letting a the first term, z= the last term, and m= the means. By the problem a=4, z=256, and m=2; therefore substituting, (256)+1=644 4 the ratio. The first term 4, therefore 4, 4X4, 4×4x4, 4x4x4×4=4, 16, 64, 256 the series. II. The first term, the last term 9, and the means =3. Therefore taking the same formula as in the first, 1 1 = () and substi. numbers, () 814—3—the ratio. And, X3, X3x3; 1x3x3x3=1, 1, 1, 3 the series. Four problems under article 442. I. By the formula, s= rz-a Putting 6 for the first term, 1458 for the last term, and 3 for the ratio, 3×1458–6 = 3-1 4374-6 4368 =2184 the sum of all the terms. 2 2 III. The last term ar"-1=(1×3)12-1=311=3 raised to the eleventh power, 177147. The sum of the terms= rz-a 3x177147-1 = 531440 =265720. Seven problems under article 444. I. Let x, y, and z= the required number. 1. By the given conditions, xy::yz 2. Converting the proportion into an equation az=y2 3. By the problem, x+y+z=14 4. " 66 66 5. Transposing in the 3d, x+z=14—y 6. Involving to the 2d power x2+2xz+z2=196-28y+y2 7. Transposing in the 4th, x2 +z2 =84—y2 8. Subtracting the 7th from the 6th, 2xz=112-28y+2y2 9. Dividing by 2, xz=56-14y+y2 10. Making the 2d and 9th equal, y2-56-14y+y2 11. Transposing and uniting, 14y=56, and y=4 |