908 CHAPTER 12 Sequences; Induction; the Binomial Theorem In Problems 6–11, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 6. 6, 12, 36, 144, . . . 7. { } − ⋅ 1 2 4n 8. − − − − 2, 10, 18, 26, . . . 9. { } − + n 2 7 10. 25, 10, 4, 8 5 , . . . 11. { } − + n n 2 3 2 1 12. Determine whether the infinite geometric series − + − + 256 64 16 4 converges or diverges. If it converges, find its sum. 13. Expand ( ) +m3 2 5 using the Binomial Theorem. 14. Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. ( )( )( ) ( ) + + + + = + n n 1 1 1 1 1 2 1 1 3 1 1 1 15. A new car sold for $31,000. If the vehicle loses 15% of its value each year, how much will it be worth after 10 years? 16. A weightlifter begins his routine by benching 100 pounds and increases the weight by 30 pounds for each set. If he does 10 repetitions in each set, what is the total weight lifted after 5 sets? Cumulative Review 1. Find all the solutions, real and complex, of the equation = x 9 2 2. (a) Graph the circle + = x y 100 2 2 and the parabola = y x3 .2 (b) Solve the system of equations: + = = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y y x 100 3 2 2 2 (c) Where do the circle and the parabola intersect? 3. Solve the equation: = e2 5 x 4. Find an equation of the line with slope 5 and x-intercept 2. 5. Find the standard equation of the circle whose center is the point ( ) −1, 2 if ( ) 3, 5 is a point on the circle. 6. ( ) = − f x x x 3 2 and ( ) = + g x x2 1 Find: (a) ( )( ) f g 2 (b) ( )( ) g f 4 (c) ( )( ) f g x (d) The domain of ( )( ) f g x (e) ( )( ) g f x (f ) The domain of ( )( ) g f x (g) The function −g 1 and its domain (h) The function −f 1 and its domain 7. Find an equation of an ellipse with center at the origin, a focus at ( ) 0, 3 , and a vertex at ( ) 0, 4 . 8. Find an equation of a parabola with vertex at ( ) −1, 2 and focus at ( ) −1, 3 . 9. Find the polar equation of a circle with center at ( ) 0, 4 that passes through the pole. What is the rectangular equation? 10. Solve the equation π − − = ≤ < x x x 2sin sin 3 0, 0 2 2 11. Find the exact value of ( ) − − cos 0.5 . 1 12. If θ = sin 1 4 and θ is in the second quadrant, find: (a) θ cos (b) θ tan (c) θ ( ) sin 2 (d) θ ( ) cos 2 (e) θ ( ) sin 1 2
RkJQdWJsaXNoZXIy NjM5ODQ=