Chapter 11

Prove that each statement is true for all positive integers. $$ 1+5+9+\cdots+(4 n-3)=n(2 n-1) $$

Evaluate each expression. $$ \frac{12 !}{2 ! 10 !} $$

Find the first three iterates of each function for the given initial value. $$ f(x)=-2 x+5, x_{0}=2 $$

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} 35\left(-\frac{3}{4}\right)^{n-1}$$

Find the sum of each geometric series. $$ \sum_{n=1}^{12} \frac{1}{6}(-2)^{n} $$

Write an equation for the \(n\) th term of the geometric sequence \(4,8,16, \ldots\)

Find the indicated term of each arithmetic sequence. \(a_{1}=3, d=-5, n=24\)

Prove that each statement is true for all positive integers. $$ 2+5+8+\cdots+(3 n-1)=\frac{n(3 n+1)}{2} $$

Find the indicated term of each expansion. fourth term of \((a+b)^{8}\)

Find the first three iterates of each function for the given initial value. $$ f(x)=x^{2}+2, x_{0}=-1 $$

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} \frac{1}{2}\left(\frac{3}{8}\right)^{n-1}$$

Find the sum of each geometric series. $$ \sum_{n=1}^{8} \frac{1}{3} \cdot 5^{n-1} $$

Write an equation for the \(n\) th term of the geometric sequence \(15,5, \frac{5}{3}, . .\)

Find the indicated term of each arithmetic sequence. \(a_{1}=-5, d=7, n=13\)

Find \(S_{n}\) for each arithmetic series described. $$ d=5, n=16, a_{n}=72 $$

Prove that each statement is true for all positive integers. $$ 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$

Find the indicated term of each expansion. fifth term of \((2 a+3 b)^{10}\)

Find the first five terms of each sequence. $$ a_{1}=-6, a_{n+1}=a_{n}+3 $$

Write each repeating decimal as a fraction. \(0 . \overline{5}\)

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

Find the indicated term of each geometric sequence. $$ a_{3}=24, r=\frac{1}{2}, n=7 $$

Find the indicated term of each arithmetic sequence. \(a_{1}=-4, d=\frac{1}{3}, n=8\)

Find \(a_{1}\) for each arithmetic series described. $$ d=8, n=19, S_{19}=1786 $$

Prove that each statement is true for all positive integers. $$ 1^{2}+3^{2}+5^{2}+\cdots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3} $$

Expand each power. $$ (a-b)^{3} $$

Find the first five terms of each sequence. $$ a_{1}=13, a_{n+1}=a_{n}+5 $$

Write each repeating decimal as a fraction. \(0 . \overline{73}\)

Find the sum of each geometric series. $$ \sum_{n=1}^{9} \frac{1}{27}(-3)^{n-1} $$

Find the indicated term of each geometric sequence. $$ a_{3}=32, r=-0.5, n=6 $$

Find the indicated term of each arithmetic sequence. \(a_{1}=6.6, d=1.05, n=32\)

Find \(a_{1}\) for each arithmetic series described. $$ d=-2, n=12, S_{12}=96 $$

Prove that each statement is true for all positive integers. \(8^{n}-1\) is divisible by 7

Expand each power. $$ (m+n)^{4} $$

Find the first five terms of each sequence. $$ a_{1}=2, a_{n+1}=a_{n}-n $$

Write each repeating decimal as a fraction. \(0 . \overline{175}\)

Find \(S_{n}\) for each geometric series described. $$ a_{1}=12, a_{5}=972, r=-3 $$

Find two geometric means between 1 and \(27 .\)

A basketball team has a halftime promotion where a fan gets to shoot a 3 -pointer to try to win a jackpot. The jackpot starts at \(\$ 5000\) for the first game and increases \(\$ 500\) each time there is no winner. Ellis has tickets to the fifteenth game of the season. How much will the jackpot be for that game if no one wins by then?

Prove that each statement is true for all positive integers. \(9^{n}-1\) is divisible by 8

Expand each power. $$ (r+s)^{8} $$

Find the first five terms of each sequence. $$ a_{1}=6, a_{n+1}=a_{n}+n+3 $$

Find the sum of each infinite geometric series, if it exists. \(a_{1}=4, r=\frac{5}{7}\)

Find two geometric means between 2 and 54

Write an equation for the \(n\) th term of the arithmetic sequence \(-26,-15\) \(-4,7, \ldots\)

ARCHITECTURE A memorial being constructed in a city park will be a brick wall, with a top row of six gold-plated bricks engraved with the names of six local war veterans. Each row has two more bricks than the row above it. Prove that the number of bricks in the top \(n\) rows is \(n^{2}+5 n\)

Expand each power. $$ (m-a)^{5} $$

Find the first five terms of each sequence. $$ a_{1}=9, a_{n+1}=2 a_{n}-4 $$

Find \(S_{n}\) for each geometric series described. $$ a_{1}=5, a_{n}=81,920, r=4 $$

Find the next two terms of each geometric sequence. $$ 405,135,45, \dots $$

Complete: 68 is the \(\underline{?}\) th term of the arithmetic sequence \(-2,3,8, \ldots\)