Chapter 9

A deposit of \(5000\)dollar is made in an account that earns \(3 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by $$A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ (a) Compute the first eight terms of this sequence. (b) Find the balance in this account after 10 years by computing the \(40\)th term of the sequence.

The expansions of \((x+y)^{4},(x+y)^{5},\) and \((x+y)^{6}\) are as follows. $$\begin{aligned} (x+y)^{4}=& 1 x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+1 y^{4} \\ (x+y)^{5}=& 1 x^{5}+5 x^{4} y+10 x^{3} y^{2}+10 x^{2} y^{3} \\ (x+y)^{6}=& 1 x^{6}+6 x^{5} y+15 x^{4}+1 y^{5} \\ & \quad+6 x y^{5}+1 y^{6} \end{aligned}$$

An investor deposits \(10,000\)dollar in an account that earns \(3.5 \%\) interest compounded monthly. The balance in the account after \(n\) months is given by $$A_{n}=10,000\left(1+\frac{0.035}{12}\right)^{n}, \quad n=1,2,3, . . .$$ (a) Write the first eight terms of the sequence. (b) Find the balance in the account after 5 years by computing the 60 th term of the sequence. (c) Is the balance after 10 years twice the balance after 5 years? Explain.

Finding a Determinant Find the determinant of the matrix. $$\left[\begin{array}{rrr} -1 & 3 & 4 \\ -2 & 8 & 0 \\ 0 & 5 & -1 \end{array}\right]$$

Prove the property for all integers \(r\) and \(n,\) where \(0 \leq r \leq n\) \(_{n} C_{r}=_{n} C_{n-r}\)

A landlocked lake has been selected to be stocked in the year 2015 with 5500 trout, and to be restocked each year thereafter with 500 trout. Each year the fish population declines \(25 \%\) due to harvesting and other natural causes. (a) Write a recursive sequence that gives the population \(p_{n}\) of trout in the lake in terms of the year \(n,\) with \(n=0\) corresponding to 2015. (b) Use the recursion formula from part (a) to find the numbers of trout in the lake for \(n=1,2,3\) and \(4 .\) Interpret these values in the context of the situation. (c) Use a graphing utility to find the number of trout in the lake as time passes infinitely. Explain your result.

Finding a Determinant Find the determinant of the matrix. $$\left[\begin{array}{rrr} -1 & 0 & 4 \\ -4 & 3 & 5 \\ 0 & 2 & -3 \end{array}\right]$$

Prove the property for all integers \(r\) and \(n,\) where \(0 \leq r \leq n\) \(_{n} C_{0}-_{n} C_{1}+_{n} C_{2}-\cdots \pm_{n} C_{n}=0\)

Determine whether the statement is true or false. Justify your answer. $$\sum_{i=1}^{4}\left(i^{2}+2 i\right)=\sum_{i=1}^{4} i^{2}+2 \sum_{i=1}^{4} i$$

The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\).

Determine whether the statement is true or false. Justify your answer. $$\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}$$

Use the Fibonacci sequence. (See Example 5.) Write the first 12 terms of the Fibonacci sequence \(a_{n}\) and the first 10 terms of the sequence given by $$b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n>0$$.

Find the inverse of the matrix. \(\left[\begin{array}{rr}-1 & -4 \\ 1 & 2\end{array}\right]\)

Find the inverse of the matrix. \(\left[\begin{array}{cc}11 & -12 \\ 2 & -2\end{array}\right]\)

Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with \(n\) th term \(a_{n}\). Use the table feature of a graphing utility to find the first five terms of the sequence.

Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with \(n\) th term \(a_{n}\). Find expressions for \(a_{n+1}\) and \(a_{n+2}\) in terms of \(n\).

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{2 n+1}}{2 n+1}$$

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{n+1}}{n+1}$$

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{2 n}}{(2 n) !}$$

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{2 n+1}}{(2 n+1) !}$$

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{n}}{n !}$$

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{n+1}}{(n+1) !}$$

Write the first five terms of the sequence. Then find an expression for the \(n\) th partial sum. $$a_{n}=\frac{1}{2 n}-\frac{1}{2 n+2}$$

Write the first five terms of the sequence. Then find an expression for the \(n\) th partial sum. $$a_{n}=\frac{1}{n}-\frac{1}{n+1}$$

Write the first five terms of the sequence. Then find an expression for the \(n\) th partial sum. $$a_{n}=\frac{1}{n+1}-\frac{1}{n+2}$$

Write the first five terms of the sequence. Then find an expression for the \(n\) th partial sum. $$a_{n}=\frac{1}{n}-\frac{1}{n+2}$$

Does every finite series whose terms are integers have a finite sum? Explain.

Find, if possible, (a) \(A-B,\) (b) \(2 B-3 A,\) (c) \(A B,\) and (d) \(B A.\) $$A=\left[\begin{array}{l}6 \\\3\end{array}\right], B=\left[\begin{array}{r}4 \\\\-3\end{array}\right]$$

Find, if possible, (a) \(A-B,\) (b) \(2 B-3 A,\) (c) \(A B,\) and (d) \(B A.\) $$A=\left[\begin{array}{cc}10 & 7 \\\\-4 & 6\end{array}\right], B=\left[\begin{array}{cc}0 & -12 \\\8 & 11\end{array}\right]$$

Find, if possible, (a) \(A-B,\) (b) \(2 B-3 A,\) (c) \(A B,\) and (d) \(B A.\) $$A=\left[\begin{array}{rrr}-2 & -3 & 6 \\\4 & 5 & 7 \\\1 & 7 & 4\end{array}\right], B=\left[\begin{array}{lll}1 & 4 & 2 \\\0 & 1 & 6 \\\0 & 3 & 1\end{array}\right]$$