Chapter 11

Find the sum of each arithmetic series. $$ \sum_{n=1}^{7}(2 n+1) $$

NATURE The terms of the Fibonacci sequence are found in many places in nature. The number of spirals of seeds in sunflowers are Fibonacci numbers, as are the number of spirals of scales on a pinecone. The Fibonacci sequence begins \(1,1,2,3,5,8, \ldots\) Each element after the first two is found by adding the previous two terms. If \(f_{n}\) stands for the \(n\) th Fibonacci number, prove that \(f_{1}+f_{2}+\ldots+f_{n}=f_{n+2}-1\)

Expand each power. $$ (x+3)^{5} $$

Find the first five terms of each sequence. $$ a_{1}=4, a_{n+1}=3 a_{n}-6 $$

Find the sum of each infinite geometric series, if it exists. \(a_{1}=12, r=-0.6\)

Find \(S_{n}\) for each geometric series described. $$ a_{1}=-8, a_{6}=-256, r=2 $$

Find the next two terms of each geometric sequence. $$ 81,108,144, \dots $$

Find the sum of each arithmetic series. $$ \sum_{k=3}^{7}(3 k+4) $$

Find the three arithmetic means between 44 and 92.

Find a counterexample for each statement. $$ 1^{2}+2^{2}+3^{2}+\dots+n^{2}=\frac{n(3 n-1)}{2} $$

Expand each power. $$ (a-2)^{4} $$

If \(a_{0}=7\) and \(a_{n+1}=a_{n}+12\) for \(n \geq 0,\) find the value of \(a_{5}\)

Find the sum of each infinite geometric series, if it exists. \(a_{1}=18, r=0.6\)

Find the indicated term for each geometric series described. $$ S_{n}=\frac{381}{64}, r=\frac{1}{2}, n=7 ; a_{1} $$

Find the three arithmetic means between 2.5 and 12.5.

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=7, a_{n}=79, n=8 $$

Evaluate each expression. $$ 9 ! $$

Find the sum of each infinite geometric series, if it exists. \(16+12+9+\cdots\)

Find the next two terms of each geometric sequence. $$ 162,108,72, \dots $$

Find the next four terms of each arithmetic sequence. \(9,16,23, \ldots\)

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=58, a_{n}=-7, n=26 $$

Find a counterexample for each statement. $$ 3^{n}+1 \text { is divisible by } 4 $$

Find the first three iterates of each function for the given initial value. $$ f(x)=9 x-2, x_{0}=2 $$

Find the sum of each infinite geometric series, if it exists. \(-8-4-2-\cdots\)

Find the indicated term for each geometric series described. $$ S_{n}=443, r=\frac{1}{3}, n=6 ; a_{1} $$

Find the first five terms of each geometric sequence described. $$ a_{1}=2, r=-3 $$

Find the next four terms of each arithmetic sequence. \(31,24,17, \dots\)

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=7, d=-2, n=9 $$

Find a counterexample for each statement. $$ 2^{n}+2 n^{2} \text { is divisible by } 4 $$

Evaluate each expression. $$ \frac{9 !}{7 !} $$

Find the first three iterates of each function for the given initial value. $$ f(x)=4 x-3, x_{0}=2 $$

Find the sum of each infinite geometric series, if it exists. \(12-18+24-\cdots\)

Find the first five terms of each geometric sequence described. $$ a_{1}=1, r=4 $$

Find the next four terms of each arithmetic sequence. \(-6,-2,2, \dots\)

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=3, d=-4, n=8 $$

Evaluate each expression. $$ \frac{7 !}{4 !} $$

Find the first three iterates of each function for the given initial value. $$ f(x)=3 x+5, x_{0}=-4 $$

Find the sum of each infinite geometric series, if it exists. \(18-12+8-\cdots\)

Find \(S_{n}\) for each geometric series described. $$ a_{1}=2, a_{6}=486, r=3 $$

Find the first five terms of each geometric sequence described. Find \(a_{7}\) if \(a_{1}=12\) and \(r=\frac{1}{2}\)

Find the next four terms of each arithmetic sequence. \(-8,-5,-2, \dots\)

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=5, d=\frac{1}{2}, n=13 $$

Find the indicated term of each expansion. sixth term of \((x-y)^{9}\)

Find the first three iterates of each function for the given initial value. $$ f(x)=5 x+1, x_{0}=-1 $$

Find the sum of each infinite geometric series, if it exists. \(1+\frac{2}{3}+\frac{4}{9}+\cdots\)

Find \(S_{n}\) for each geometric series described. $$ a_{1}=3, a_{8}=384, r=2 $$

Find the first five terms of each geometric sequence described. Find \(a_{6}\) if \(a_{1}=\frac{1}{3}\) and \(r=6\)

Find the first five terms of each arithmetic sequence described. \(a_{1}=2, d=13\)

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=12, d=\frac{1}{3}, n=13 $$

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