91Ó°ÊÓ

Describe the similarities and differences between arithmetic sequences and linear functions.

Problems \(75-84\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If a credit card charges \(15.3 \%\) interest compounded monthly, find the effective rate of interest.

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ 2,4,6,8, \ldots $$

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{\left(\frac{5}{4}\right)^{n}\right\\} $$

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ -1,2,-4,8, \ldots $$

If you have been hired at an annual salary of \(\$ 42,000\) and expect to receive annual increases of \(3 \%,\) what will your salary be when you begin your fifth year?

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: $$ a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2} $$ where \(n \geq 2\) is the number of the planet from the sun. (a) Determine the first eight terms of the sequence. (b) At the time of Bode's publication, the known planets were Mercury \((0.39 \mathrm{AU}),\) Venus \((0.72 \mathrm{AU}),\) Earth \((1 \mathrm{AU})\) Mars \((1.52 \mathrm{AU}),\) Jupiter \((5.20 \mathrm{AU}),\) and Saturn \((9.54 \mathrm{AU})\) 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 \(19.2 \mathrm{AU}\) and \(2.77 \mathrm{AU},\) 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 \(30.07 \mathrm{AU}\) and \(39.44 \mathrm{AU},\) 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.

Don contributes \(\$ 500\) at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80 th deposit ( 20 years) if the per annum rate of return is assumed to be \(5 \%\) compounded quarterly?

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 \(I_{0}\) before it falls on a mirror, write the \(n\) th 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 \(98 \% ?\)

\(\sqrt{8}\)