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Chapter 14: Problem 31
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$1^{2}+2^{2}+3^{2}+\cdots+15^{2}$$
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Use the formula for the sum of an infinite geometric series to solve Exercises. A new factory in a small town has an annual payroll of \(\$ 6\) million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?
Use the formula for the sum of the first n terms of a geometric sequence to solve. A pendulum swings through an arc of 20 inches. On each successive swing, the length of the arc is \(90 \%\) of the previous length. $$\begin{aligned}&20, \quad\quad 0.9(20), \quad0.9^{2}(20), \quad 0.9^{3}(20), \ldots\\\&\begin{array}{|c|c|c|c|}\hline \text { 1st } & \text { 2nd } & \text { 3rd } & \text { 4th } \\\\\text { swing } & \text { swing } & \text { swing } & \text { swing } \\\\\hline\end{array}\end{aligned}$$ After 10 swings, what is the total length of the distance the pendulum has swung? Round to the nearest hundredth of an inch.
A deposit of 10,000 dollars is made in an account that earns \(8 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}=10,000\left(1+\frac{0.08}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ Find the balance in the account after six years. Round to the nearest cent.
Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the exponents on \(b\).
Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$\frac{4}{1}, \frac{9}{2}, \frac{16}{3}, \frac{25}{4}, \dots$$
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