13. The 9th term of an arithmetical progression is 1/6; the 16th term is 5/2. Find the first term. (_Regents._) Solve graphically: 1. x^2 - x - 6 = 0. 2. x^2 + 3x - 10 = 0. 3. Find four numbers in arithmetical progression, such that the sum of the first two is 1, and the sum of the last two is -19. 4. What number added to 2, 20, 9, 34, will make the results proportional? 5. Find the middle term of [3a^5 + (b^(3/4))/(2)]^8. 6. Solve (x + 1)/(3x + 2) = (2x - 3)/(3x - 2) - 1 - 36/(4 - 9x^2). (_Princeton._) 7. A strip of carpet one half inch thick and 29-6/7 feet long is rolled on a roller four inches in diameter. Find how many turns there will be, remembering that each turn increases the diameter by one inch, and that the circumference of a circle equals (approximately) 22/7 times the diameter. (_Harvard._) 8. The sum of the first three terms of a geometrical progression is 21, and the sum of their squares is 189. What is the first term? (_Yale._) 9. Find the geometrical progression whose sum to infinity is 4, and whose second term is 3/4. 10. Solve 4x + 4[3x^2 - 7x + 3]^(1/2) = 3x^2 - 3x + 6. 11. Solve { 2x^2 + 3xy - 5y^2 = 4, { 2xy + 3y^2 = -3. 12. Two hundred stones are placed on the ground 3 feet apart, the first being 3 feet from a basket. If the basket and all the stones are in a straight line, how far does a person travel who starts from the basket and brings the stones to it one by one? Solve graphically; and check by solving algebraically: 1. { x^2 + y^2 = 25, { x + y = 1. 2. x^2 - 3x - 18 = 0. 3. x^2 + 3x - 10 = 0. Determine the value of m for which the roots of the equation will be equal: (HINT: See page 40. To have the roots equal, b^2 - 4ac must equal 0.) 4. 2x^2 - mx + 12-1/2 = 0. 5. (m - 1)x^2 + mx + 2m - 3 = 0. 6. If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other root without solving the equation. (_Univ. of Penn._) 7. How many times does a common clock strike in 12 hours? 8. Find the sum to infinity of 2/(2^(1/2)), 1/(2^(1/2)), 1/(2[2]^(1/2)), .... 9. Solve [x/2 + 6/x]^2 - 6[x/2 + 6/x] + 8 = 0. 10. Find the value of the recurring decimal 2.214214.... 11. A man purchases a $500 piano by paying monthly installments of $10 and interest on the debt. If the yearly rate is 6%, what is the total amount of interest? 12. The arithmetical mean between two numbers is 42-1/2, and their geometrical mean is 42. Find the numbers. (_College Entrance Exam. Board._)