Chapter 13

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{f_{n}\right\\}=\left\\{3^{2 n}\right\\} $$

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\left\\{n^{2}+1\right\\}\)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{1}{3} n(n+1)(n+2) $$

Expand each expression using the Binomial Theorem. $$ (x+1)^{5} $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{t_{n}\right\\}=\left\\{\frac{3^{n-1}}{2^{n}}\right\\} $$

List the first five terms of each sequence. \(\left\\{a_{n}\right\\}=\left\\{\frac{n}{n+2}\right\\}\)

In Problems 17-24, find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=2 ; \quad d=3 $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1 \cdot 2+3 \cdot 4+5 \cdot 6+\cdots+(2 n-1)(2 n)=\frac{1}{3} n(n+1)(4 n-1) $$

Expand each expression using the Binomial Theorem. $$ (x-1)^{5} $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{u_{n}\right\\}=\left\\{\frac{2^{n}}{3^{n-1}}\right\\} $$

List the first five terms of each sequence. \(\left\\{b_{n}\right\\}=\left\\{\frac{2 n+1}{2 n}\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=-2 ; \quad d=4 $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n^{2}+n \text { is divisible by } 2 $$

Expand each expression using the Binomial Theorem. $$ (x-2)^{6} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=2 ; \quad r=3 $$

List the first five terms of each sequence. \(\left\\{c_{n}\right\\}=\left\\{(-1)^{n+1} n^{2}\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=8 ; \quad d=-7 $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n^{3}+2 n \text { is divisible by } 3 $$

Expand each expression using the Binomial Theorem. $$ (x+3)^{5} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=-2 ; \quad r=4 $$

List the first five terms of each sequence. \(\left\\{d_{n}\right\\}=\left\\{(-1)^{n-1}\left(\frac{n}{2 n-1}\right)\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=6 ; \quad d=-2 $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n^{2}-n+2 \text { is divisible by } 2 $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=5 ; \quad r=-1 $$

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\left\\{\frac{3^{n}}{2^{n}+3}\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=0 ; \quad d=\frac{1}{2} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n(n+1)(n+2) \text { is divisible by } 6 $$

Expand each expression using the Binomial Theorem. $$ (2 x+3)^{5} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=6 ; \quad r=-2 $$

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\left\\{\left(\frac{4}{3}\right)^{n}\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=1 ; \quad d=-\frac{1}{3} $$

Expand each expression using the Binomial Theorem. $$ \left(x^{2}+y^{2}\right)^{5} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=0 ; \quad r=\frac{1}{7} $$

List the first five terms of each sequence. \(\left\\{t_{n}\right\\}=\left\\{\frac{(-1)^{n}}{(n+1)(n+2)}\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=\sqrt{2} ; \quad d=\sqrt{2} $$

Prove each statement. $$ \text { If } 0

Expand each expression using the Binomial Theorem. $$ \left(x^{2}-y^{2}\right)^{6} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=1 ; \quad r=-\frac{1}{3} $$

List the first five terms of each sequence. \(\left\\{a_{n}\right\\}=\left\\{\frac{3^{n}}{n}\right\\}\)

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=0 ; \quad d=\pi $$

Prove each statement. $$ \begin{aligned} &a-b \text { is a factor of } a^{n}-b^{n}\\\ &\text { [Hint: } \left.a^{k+1}-b^{k+1}=a\left(a^{k}-b^{k}\right)+b^{k}(a-b)\right] \end{aligned} $$

Expand each expression using the Binomial Theorem. $$ (\sqrt{x}+\sqrt{2})^{6} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=\sqrt{3} ; \quad r=\sqrt{3} $$

In Problems 25-30, find the indicated term in each arithmetic sequence. $$ \text { 100th term of } 2,4,6, \ldots $$

List the first five terms of each sequence. \(\left\\{b_{n}\right\\}=\left\\{\frac{n}{e^{n}}\right\\}\)

Expand each expression using the Binomial Theorem. $$ (\sqrt{x}-\sqrt{3})^{4} $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=0 ; \quad r=\frac{1}{\pi} $$

Find the indicated term in each arithmetic sequence. $$ \text { 80th term of }-1,1,3, \ldots $$

List the first five terms of each sequence. \(\left\\{c_{n}\right\\}=\left\\{\frac{n^{2}}{2^{n}}\right\\}\)

Expand each expression using the Binomial Theorem. $$ (a x+b y)^{5} $$