Bode's Law In $1772,$ Johann Bode published the following formula for predicting the mean distances, in astronomical units $(\mathrm{AU}),$ of the planets from the sun: $$ a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2} $$ where $n \geq 2$ is the number of the planet from the sun. $$ \begin{array}{l}{\text { (a) Determine the first eight terms of this sequence. }} \\ {\text { (b) At the time of Bode's publication, the known planets were }} \\ {\text { Mercury }(0.39 \mathrm{AU}), \text { Venus }(0.72 \mathrm{AU}), \text { Earth }(1 \mathrm{AU}) \text { , Mars }} \\ {(1.52 \mathrm{AU}) \text { , Jupiter }(5.20 \mathrm{AU}), \text { and Saturn }(9.54 \mathrm{AU}) \text { . How do }} \\ {\text { the actual distances compare to the terms of the sequence? }}\end{array} $$$$ \begin{array}{l}{\text { (c) The planet Uranus was discovered in } 1781, \text { and the }} \\ {\text { asteroid Ceres was discovered in } 1801 \text { . The mean orbital }} \\ {\text { distances from the sun to Uranus and Ceres' are } 19.2} \\ {\text { AU and } 2.77 \text { AU, respectively. How well do these values }} \\ {\text { fit within the sequence? }} \\ {\text { (d) Determine the ninth and tenth terms of Bode's sequence. }}\end{array} $$$$ \begin{array}{l}{\text { (e) The planets Neptune and Pluto* were discovered in }} \\ {1846 \text { and } 1930 \text { , respectively. Their mean orbital distances }} \\ {\text { from the sun are } 30.07 \mathrm{AU} \text { and } 39.44 \mathrm{AU} \text { , respectively. }} \\ {\text { How do these actual distances compare to the terms of }} \\ {\text { the sequence? }}\end{array} $$$$ \begin{array}{l}{\text { (f) On July } 29,2005, \text { NASA announced the discovery of a }} \\ {\text { dwarf planet" }(n=11), \text { which has been named Eris Use }} \\ {\text { Bode's Law to predict the mean orbital distance of Eris }} \\ {\text { from the sun. Its actual mean distance is not yet known, but }} \\ {\text { Eris is currently about } 97 \text { astronomical units from the sun. }}\end{array} $$