Get started today!

Good to have you back!
If you've signed in to StudyBlue with Facebook in the past, please do that again.

Anonymous

Math 618 Answer to Problem 2.3.11 Autumn 2005 Algebra/Calculus Reminder Let Sn = (Ia)n| = v + 2v2 + 3v3 + ··· + nvn. We have the formula: Sn = (Ia)n| = v + 2v2 + 3v3 + ··· + nvn = ¨an| − nv n i . Using ¨an| = 1 + v + v2 + ··· + vn−1 = 1 − v n 1 − v in the above formula, we have Sn = (Ia)n| = 1−vn 1−v − nv n i . If |v| < 1, then limn→∞vn = 0 and limn→∞nvn = 0 (why?). Let n → 0 in the formula of Sn above, we get the increasing perprtuity: S∞ = v + 2v2 + 3v3 + 4v4 + ··· = (Ia)∞| = parenleftbigg 1 1 − v parenrightbigg /i = 1i + 1i2. (1) We also have a∞| = v + v2 + v3 + ··· + vk + ··· = 1i, (2) and ¨a∞| = 1 + v + v2 + ··· + vk + ··· = 11 − v = 1 + ii . (3) Solution Sandy’s perpetuity can be explained by the following diagram (v = 11+i): . . . . . . . . .0 1 2 3 k 90 + 10 90 + 20 90 + 30 . . . 90 + 10k . . . (90 + 10)v (90 + 20)v2 (90 + 30)v3 . . . (90 + 10k)vk . . . a27 a27 a27 a27 a27 a27 The sum of these present values is (using (1) and (2)): (90 + 10)v + (90 + 20)v2 + (90 + 30)v3 + ··· + (90 + 10k)vk + ··· = (90v + 90v2 + 90v3 + ··· + 90vk + ··· ) + (10v + 20v2 + 30v3 + ··· + 10kvk + ··· ) = 90a∞| + 10(Ia)∞| = 901i + 10 parenleftbigg1 i + 1 i2 parenrightbigg = 100i + 10i2 (4) Similarly, Danny’s perpetuity-due can be explained by the following diagram: . . . . . . . . .0 1 2 k 180 180 . . . 180 . . . 180 180v 180v2 . . . 180vk . . . a27 a27 a27 a27 a27 The sum of these present values is: 180 + 180v + 180v2 + ··· + 180vk + ··· = 180¨a∞| = 1801 + ii . (5) Since the two present values in (4) and (5) are equal, we have 100 i + 10 i2 = 180 1 + i i . Simplifying the last equation, we get a quadratic equation 18i2 +8i−1 = 0. Solving this equation (note that i > 0), we get i = 0.10171955. Answer: i = 0.1017

Advertisement
)

Advertisement

"StudyBlue is great for studying. I love the study guides, flashcards and quizzes. So extremely helpful for all of my classes!"

Alice , Arizona State University"I'm a student using StudyBlue, and I can 100% say that it helps me so much. Study materials for almost every subject in school are available in StudyBlue. It is so helpful for my education!"

Tim , University of Florida"StudyBlue provides way more features than other studying apps, and thus allows me to learn very quickly!??I actually feel much more comfortable taking my exams after I study with this app. It's amazing!"

Jennifer , Rutgers University"I love flashcards but carrying around physical flashcards is cumbersome and simply outdated. StudyBlue is exactly what I was looking for!"

Justin , LSU