868 CHAPTER 12 Sequences; Induction; the Binomial Theorem 99. Reflections in a Mirror A highly reflective mirror reflects 95% of the light that falls on it. In a light box having walls made of the mirror, the light reflects back-and-forth between the mirrors. (a) If the original intensity of the light is I0 before it falls on a mirror, write the nth term of the sequence that describes the intensity of the light after n reflections. (b) How many reflections are needed to reduce the light intensity by at least 98%? 100. Show that ( ) ( ) + + + − + = + n n n n 1 2 1 1 2 [Hint: Let ( ) ( ) ( ) = + + + − + = + − + − + + S n n S n n n 1 2 1 1 2 1 Add these equations. Then [ ] ( ) [ ] [ ] = + + + − + + + nterms in bracket S n n n 2 1 2 1 . . . 1 Now complete the derivation.] Computing Square Roots A method for approximating p can be traced back to the Babylonians. The formula is given by the recursively defined sequence = = + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − − a k a a p a 1 2 n n n 0 1 1 where k is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding a .5 Compare this result to the value provided by your calculator. 101. 5 102. 8 103. 21 104. 89 105. Triangular Numbers A triangular number is a term of the sequence ( ) = = + + + u u u n 1 1 n n 1 1 List the first seven triangular numbers. 106. Challenge Problem For the sequence given in Problem 105, show that ( )( ) = + + + u n n 1 2 2 . n 1 107. Challenge Problem For the sequence given in Problem 105, show that ( ) + = + + u u n 1 n n 1 2 108. Challenge Problem If the terms of a sequence have the property that = = = − a a a a a a , n n 1 2 2 3 1 show that = + a a a a . n n n 1 2 1 1 [Hint: Let r equal the common ratio so = = = = − a a a a a a r. n n 1 2 2 3 1 ] 97. Bode’s Law In 1772, Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun: = = + ⋅ − a a 0.4 0.4 0.3 2 n n 1 2 where ≥ n 2 is the number of the planet from the sun. (a) Determine the first eight terms of this sequence. (b) At the time of Bode’s publication, the known planets were Mercury (0.39 AU), Venus (0.72 AU), Earth (1 AU), Mars (1.52 AU), Jupiter (5.20 AU), and Saturn (9.54 AU). How do the actual distances compare to the terms of the sequence? (c) The planet Uranus was discovered in 1781, and the asteroid Ceres was discovered in 1801. The mean orbital distances from the sun to Uranus and Ceres* are 19.2 AU and 2.77 AU, respectively. How well do these values fit within the sequence? (d) Determine the ninth and tenth terms of Bode’s sequence. (e) The planets Neptune and Pluto* were discovered in 1846 and 1930, respectively.Their mean orbital distances from the sun are 30.07 AU and 39.44 AU, respectively. How do these actual distances compare to the terms of the sequence? (f) On July 29, 2005, NASA announced the discovery of a dwarf planet ( ) = n 11 , which has been named Eris.* Use Bode’s Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun. Source: NASA 98. Droste Effect The Droste Effect, named after the image on boxes of Droste cocoa powder, refers to an image that contains within it a smaller version of the image, which in turn contains an even smaller version, and so on. If each version of the image is 1 5 the height of the previous version, the height of the nth version is given by = − a a 1 5 . n n 1 Suppose a Droste image on a package has a height of 4 inches. How tall would the image be in the 6th version? *Ceres, Haumea, Makemake, Pluto, and Eris are referred to as dwarf planets. Credit: frankie’s/Shutterstock
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