Chapter 13

Find the indicated term of each geometric sequence. 7th term of \(1, \frac{1}{2}, \frac{1}{4}\)

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\)

Find the indicated term in each arithmetic sequence. $$ \text { 90th term of } 3,-3,-9, \ldots $$

Show that the statement \(" n^{2}-n+41\) is a prime number" is true for \(n=1\) but is not true for \(n=41\).

Expand each expression using the Binomial Theorem. $$ (a x-b y)^{4} $$

Find the indicated term of each geometric sequence. 8th term of \(1,3,9, \ldots\)

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \ldots\)

Find the indicated term in each arithmetic sequence. $$ 80 \text { th term of } 5,0,-5, \ldots $$

Find the indicated term of each geometric sequence. 15th term of \(1,-1,1, \ldots\)

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{6}\) in the expansion of \((x+3)^{10}\)

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\)

Find the indicated term in each arithmetic sequence. $$ \text { 80th term of } 2, \frac{5}{2}, 3, \frac{7}{2}, \ldots $$

Use mathematical induction to prove that if \(r \neq 1,\) then $$ a+a r+a r^{2}+\cdots+a r^{n-1}=a \frac{1-r^{n}}{1-r} $$

Find the indicated term of each geometric sequence. 10th term of \(-1,2,-4, \ldots\)

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{3}\) in the expansion of \((x-3)^{10}\)

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots\)

Find the indicated term in each arithmetic sequence. $$ \text { 70th term of } 2 \sqrt{5}, 4 \sqrt{5}, 6 \sqrt{5}, \ldots $$

Use mathematical induction to prove that $$ \begin{aligned} a+(a+d)+(a+2 d) & \\ +\cdots+[a+(n-1) d] &=n a+d \frac{n(n-1)}{2} \end{aligned} $$

Find the indicated term of each geometric sequence. 8th term of \(0.4,0.04,0.004, \ldots\)

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{7}\) in the expansion of \((2 x-1)^{12}\)

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(1,-1,1,-1,1,-1, \ldots\)

In Problems \(31-38\), find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 8th term is \(8 ; 20\) th term is 44

Extended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions I and II hold, that is, (I) A statement is true for a natural number \(j\). (II) If the statement is true for some natural number \(k \geq j\), then it is also true for the next natural number \(k+1\). then the statement is true for all natural numbers \(\geq j\). Use the Extended Principle of Mathematical Induction to show that the number of diagonals in a convex polygon of \(n\) sides is \(\frac{1}{2} n(n-3)\) [Hint: Begin by showing that the result is true when \(n=4\) (Condition I).]

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{3}\) in the expansion of \((2 x+1)^{12}\)

Find the indicated term of each geometric sequence. 7th term of \(0.1,1.0,10.0, \ldots\)

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 4th term is \(3 ; 20\) th term is 35

Geometry Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of \(n\) sides equals \((n-2) \cdot 180^{\circ} .\)

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 9th term is \(-5 ; 15\) th term is 31

Challenge Problem Use the Principle of Mathematical Induction to prove that $$ \left[\begin{array}{rr} 5 & -8 \\ 2 & -3 \end{array}\right]^{n}=\left[\begin{array}{cr} 4 n+1 & -8 n \\ 2 n & 1-4 n \end{array}\right] $$ for all natural numbers \(n\).

Find the nth term \(a_{n}\) of each geometric sequence. When given, \(r\) is the common ratio. $$ 5,10,20,40, \ldots $$

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{2}\) in the expansion of \((2 x-3)^{9}\)

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 8th term is 4; 18th term is - 96

Challenge Problem Paper Creases If a sheet of paper is folded in half by folding the top edge down to the bottom edge, one crease will result. If the folded paper is folded in the same manner, the result is three creases. With each fold, the number of creases can be defined recursively by \(c_{1}=1, c_{n+1}=2 c_{n}+1\) (a) Find the number of creases for \(n=3\) and \(n=4\) folds. (b) Use the given information and your results from part (a) to find a formula for the number of creases after \(n\) folds, \(c_{n}\), in terms of the number of folds alone. (c) Use the Principle of Mathematical Induction to prove that the formula found in part (b) is correct for all natural numbers. (d) Tosa Tengujo is reportedly the world's thinnest paper with a thickness of \(0.02 \mathrm{~mm}\). If a piece of this paper could be folded 25 times, how tall would the stack be?

Use the Binomial Theorem to find the indicated coefficient or term. The 5 th term in the expansion of \((x+3)^{7}\)

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

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 15th term is 0 ; 40th term is -50

Find the nth term \(a_{n}\) of each geometric sequence. When given, \(r\) is the common ratio. $$ 4,1, \frac{1}{4}, \frac{1}{16}, \ldots $$

Use the Binomial Theorem to find the indicated coefficient or term. The 3 rd term in the expansion of \((x-3)^{7}\)

A sequence is defined recursively. List the first five terms. $$a_{1}=3 ; \quad a_{n}=4-a_{n-1}$$

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 5th term is \(-2 ; 13\) th term is 30

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: \(\log _{2} \sqrt{x+5}=4\)

Use the Binomial Theorem to find the indicated coefficient or term. The 3 rd term in the expansion of \((3 x-2)^{9}\)

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

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 14th term is \(-1 ; 18\) th term is -9

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 the system: \(\left\\{\begin{array}{l}4 x+3 y=-7 \\ 2 x-5 y=16\end{array}\right.\)

Find the nth term \(a_{n}\) of each geometric sequence. When given, \(r\) is the common ratio. $$ a_{2}=7 ; \quad r=\frac{1}{3} $$

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

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{0}\) in the expansion of \(\left(x^{2}+\frac{1}{x}\right)^{12}\)

A sequence is defined recursively. List the first five terms. \(a_{1}=4 ; \quad a_{n}=3 a_{n-1}\)

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. For \(A=\left[\begin{array}{rrr}1 & 2 & -1 \\ 0 & 1 & 4\end{array}\right]\) and \(B=\left[\begin{array}{rr}3 & -1 \\ 1 & 0 \\ -2 & 2\end{array}\right],\) find \(A \cdot B\)