Chapter 13

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{0}\) in the expansion of \(\left(x-\frac{1}{x^{2}}\right)^{9}\)

A sequence is defined recursively. List the first five terms. \(a_{1}=2 ; \quad a_{n}=-a_{n-1}\)

A sequence is defined recursively. List the first five terms. \(a_{1}=3 ; \quad a_{n}=\frac{a_{n-1}}{n}\)

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{2}\) in the expansion of \(\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right)^{8}\)

A sequence is defined recursively. List the first five terms. \(a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}\)

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. Solve: \(e^{3 x-7}=4\)

Use the Binomial Theorem to find the numerical value of \((1.001)^{5}\) correct to five decimal places. \(\left[\right.\) Hint: \(\left.(1.001)^{5}=\left(1+10^{-3}\right)^{5}\right]\)

A sequence is defined recursively. List the first five terms. \(a_{1}=-1 ; \quad a_{2}=1 ; \quad a_{n}=a_{n-2}+n a_{n-1}\)

Show that \(\left(\begin{array}{c}n \\ n-1\end{array}\right)=n\) and \(\left(\begin{array}{l}n \\ n\end{array}\right)=1\).

An approximation for \(n !,\) when \(n\) is large, is given by $$ n ! \approx \sqrt{2 n \pi}\left(\frac{n}{e}\right)^{n}\left(1+\frac{1}{12 n-1}\right) $$ Calculate \(12 !, 20 !,\) and \(25 !\) on your calculator. Then use Stirling's formula to approximate \(12 !, 20 !,\) and \(25 !\)

If \(n\) is a positive integer, show that \(\left(\begin{array}{l}n \\\ 0\end{array}\right)+\left(\begin{array}{l}n \\\ 1\end{array}\right)+\cdots+\left(\begin{array}{l}n \\\ n\end{array}\right)=2^{n}\) [Hint: \(2^{n}=(1+1)^{n} ;\) now use the Binomial Theorem.]

A sequence is defined recursively. List the first five terms. \(a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{2+a_{n-1}}\)

If \(n\) is a positive integer, show that \(\left(\begin{array}{l}n \\\ 0\end{array}\right)-\left(\begin{array}{l}n \\\ 1\end{array}\right)+\left(\begin{array}{l}n \\\ 2\end{array}\right)-\cdots+(-1)^{n}\left(\begin{array}{l}n \\\ n\end{array}\right)=0\)

A sequence is defined recursively. List the first five terms. \(a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{\frac{a_{n-1}}{2}}\)

Use a graphing utility to find the sum of each geometric sequence. $$ \sum_{n=1}^{15}\left(\frac{2}{3}\right)^{n} $$

Find the value of \(\left(\begin{array}{l}5 \\\ 0\end{array}\right)\left(\frac{1}{4}\right)^{5}+\left(\begin{array}{l}5 \\\ 1\end{array}\right)\left(\frac{1}{4}\right)^{4}\left(\frac{3}{4}\right)+\left(\begin{array}{l}5 \\\ 2\end{array}\right)\left(\frac{1}{4}\right)^{3}\left(\frac{3}{4}\right)^{2}\) \(+\left(\begin{array}{l}5 \\\ 3\end{array}\right)\left(\frac{1}{4}\right)^{2}\left(\frac{3}{4}\right)^{3}+\left(\begin{array}{l}5 \\\ 4\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^{4}+\left(\begin{array}{l}5 \\\ 5\end{array}\right)\left(\frac{3}{4}\right)^{5}\)

Expand each sum. \(\sum_{k=1}^{n}(k+2)\)

Use a graphing utility to find the sum of each geometric sequence. $$ \sum_{n=1}^{15} 4 \cdot 3^{n-1} $$

The entries in the Pascal Triangle can, for \(n \geq 2\), be used to determine the number of \(k\) -sided figures that can be formed using a set of \(n\) points on a circle. In general, the first entry in a row indicates the number of \(n\) -sided figures that can be formed, the second entry indicates the number of \((n-1)\) -sided figures, and so on. For example, if a circle contains 4 points, the row for \(n=4\) in the Pascal Triangle shows the number of possible quadrilaterals (1), the number of triangles (4), and the number of line segments (6) that can be formed using the four points. (a) How many hexagons can be formed using 8 points lying on the circumference of a circle? (b) How many triangles can be formed using 10 points lying on the circumference of a circle? (c) How many dodecagons can be formed using 20 points lying on the circumference of a circle?

Expand each sum. \(\sum_{k=1}^{n}(2 k+1)\)

Find the coefficient of \(x^{4}\) in \(f(x)=\left(1-x^{2}\right)+\left(1-x^{2}\right)^{2}+\cdots+\left(1-x^{2}\right)^{10}\).

Find each sum. $$ \sum_{n=1}^{80}(4 n-9) $$

In the expansion of \(\left[a+(b+c)^{2}\right]^{8}\) find the coefficient of the term containing \(a^{5} b^{4} c^{2}\).

Expand each sum. \(\sum_{k=1}^{n}(k+1)^{2}\)

Find each sum. $$ \sum_{n=1}^{90}(3-2 n) $$

Solve \(6^{x}=5^{x+1}\). Express the answer both in exact form and as a decimal rounded to three decimal places.

Expand each sum. \(\sum_{k=0}^{n} \frac{1}{3^{k}}\)

Find each sum. $$ \sum_{n=1}^{100}\left(6-\frac{1}{2} n\right) $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 2+\frac{4}{3}+\frac{8}{9}+\cdots $$

For \(\mathbf{v}=2 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{w}=3 \mathbf{i}-2 \mathbf{j}\) (a) Find the dot product \(\mathbf{v} \cdot \mathbf{w}\). (b) Find the angle between \(\mathbf{v}\) and \(\mathbf{w}\). (c) Are the vectors parallel, orthogonal, or neither?

Expand each sum. \(\sum_{k=0}^{n}\left(\frac{3}{2}\right)^{k}\)

Find each sum. $$ \sum_{n=1}^{80}\left(\frac{1}{3} n+\frac{1}{2}\right) $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 8+4+2+\cdots $$

Solve the system of equations: \(\left\\{\begin{array}{c}x-y-z=0 \\ 2 x+y+3 z=-1 \\ 4 x+2 y-z=12\end{array}\right.\)

Expand each sum. \(\sum_{k=0}^{n-1} \frac{1}{3^{k+1}}\)

Find each sum. The sum of the first 120 terms of the sequence $$ 14,16,18,20, \ldots $$

Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points. $$ \left\\{\begin{array}{r} x \geq 0 \\ y \geq 0 \\ x+y \leq 6 \\ 2 x+y \leq 10 \end{array}\right. $$

Expand each sum. \(\sum_{k=0}^{n-1}(2 k+1)\)

Find each sum. The sum of the first 46 terms of the sequence $$ 2,-1,-4,-7, \ldots $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\cdots $$

If \(f(x)=x^{2}-6\) and \(g(x)=\sqrt{x+2},\) find \(g(f(x))\) and state its domain.

Find \(x\) so that \(x+3,2 x+1,\) and \(5 x+2\) are consecutive terms of an arithmetic sequence.

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 1-\frac{3}{4}+\frac{9}{16}-\frac{27}{64}+\cdots $$

If \(y=\frac{5}{3} x^{3}+2 x+C\) and \(y=5\) when \(x=3,\) find the value of \(C\).

Expand each sum. \(\sum_{k=3}^{n}(-1)^{k+1} 2^{k}\)

Find \(x\) so that \(2 x, 3 x+2,\) and \(5 x+3\) are consecutive terms of an arithmetic sequence.

Express each sum using summation notation. \(1+2+3+\cdots+20\)

How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to obtain a sum of \(1092 ?\)

Express each sum using summation notation. \(1^{3}+2^{3}+3^{3}+\cdots+8^{3}\)

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 5\left(\frac{1}{4}\right)^{k-1} $$