Chapter 11

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

Expand each power. $$ (p+q)^{5} $$

Find the first five terms of each sequence. $$ a_{1}=12, a_{n+1}=a_{n}-3 $$

Find the sum of each infinite geometric series, if it exists. \(a_{1}=36, r=\frac{2}{3}\)

Find \(S_{n}\) for each geometric series described. $$ a_{1}=5, r=2, n=14 $$

Find the next four terms of each arithmetic sequence. \(12,16,20, \dots\)

Find the sum of each arithmetic series. $$ 5+11+17+\dots+95 $$

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

Expand each power. $$ (t+2)^{6} $$

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

Find the sum of each infinite geometric series, if it exists. \(a_{1}=18, r=-1.5\)

Find \(S_{n}\) for each geometric series described. $$ a_{1}=243, r=-\frac{2}{3}, n=5 $$

Find the first five terms of the geometric sequence for which \(a_{1}=-2\) and \(r=3 .\)

Find the sum of each arithmetic series. $$ 12+17+22+\dots+102 $$

Find the next four terms of each arithmetic sequence. \(3,1,-1, \dots\)

PARTIES Suppose that each time a new guest arrives at a party, he or she shakes hands with each person already at the party. Prove that after \(n\) guests have arrived, a total of \(\frac{n(n-1)}{2}\) handshakes have taken place.

Expand each power. $$ (x-3 y)^{4} $$

Find the first five terms of each sequence. $$ a_{1}=0, a_{n+1}=-2 a_{n}-4 $$

Find the sum of each geometric series. \(54+36+24+16+\cdots\) to 6 terms

STANDARDIZED TEST PRACTICE What is the missing term in the geometric sequence: \(-\frac{1}{4}, \frac{1}{2},-1,2,-?\) \(\mathbf{A}-4\) \(\mathbf{B}-3 \frac{1}{2}\) \(\mathrm{C} 3 \frac{1}{2}\) \(\mathbf{D} 4\)

Find the sum of each arithmetic series. $$ 38+35+32+\dots+2 $$

Find the first five terms of each arithmetic sequence described. \(a_{1}=5, d=3\)

Prove that each statement is true for all positive integers. \(4^{n}-1\) is divisible by 3

Find the first five terms of each sequence. $$ a_{1}=1, a_{2}=2, a_{n+2}=4 a_{n+1}-3 a_{n} $$

Find \(a_{9}\) for the geometric sequence \(60,30,15, \ldots\)

Find the sum of each arithmetic series. $$ 101+90+79+\dots+2 $$

Find the first five terms of each arithmetic sequence described. \(a_{1}=14, d=-2\)

Prove that each statement is true for all positive integers. \(5^{n}+3\) is divisible by 4

Evaluate each expression. $$ 8 ! $$

BANKING For Exercises 5 and \(6,\) use the following information. Rita has deposited 51000 in a bank account. At the end of each year, the bank posts 3\(\%\) interest to her account, but then takes out a \(\$ 10\) annual fee. Let \(b\) be the amount Rita deposited. Write a recursive equation for the balance \(b_{n}\) in her account at the end of \(n\) years.

WEATHER Heavy rain caused a river to rise. The river rose three inches the first day, and each day it rose twice as much as the previous day. How much did the river rise in five days?

Find \(a_{8}\) for the geometric sequence \(\frac{1}{8}, \frac{1}{4}, \frac{1}{2}, \ldots\)

TRAINING To train for a race, Rosmaria runs 1.5 hours longer each week than she did the previous week. In the first week, Rosmaria ran 3 hours. How much time will Rosmaria spend running if she trains for 12 weeks?

Find the first five terms of each arithmetic sequence described. \(a_{1}=\frac{1}{2}, d=\frac{1}{4}\)

Find a counterexample for each statement. $$ 1+2+3+\cdots+n=n^{2} $$

Evaluate each expression. $$ 10 ! $$

BANKING For Exercises 5 and \(6,\) use the following information. Rita has deposited 51000 in a bank account. At the end of each year, the bank posts 3\(\%\) interest to her account, but then takes out a \(\$ 10\) annual fee. Find the balance in the account after four years.

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} 6(-0.4)^{n-1}$$

Find the sum of each geometric series. $$ \sum_{n=1}^{5} \frac{1}{4} \cdot 2^{n-1} $$

Find the indicated term of each geometric sequence. $$ a_{1}=7, r=2, n=4 $$

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=4, a_{n}=100, n=25 $$

Find the first five terms of each arithmetic sequence described. \(a_{1}=0.5, d=-0.2\)

Find a counterexample for each statement. $$ 2^{n}+3^{n} \text { is divisible by } 4 $$

Evaluate each expression. $$ \frac{13 !}{9 !} $$

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

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} 40\left(\frac{3}{5}\right)^{n-1}$$

Find the sum of each geometric series. $$ \sum_{n=1}^{7} 81\left(\frac{1}{3}\right)^{n-1} $$

Find the indicated term of each geometric sequence. $$ a_{1}=3, r=\frac{1}{3}, n=5 $$

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=40, n=20, d=-3 $$

Find \(a_{13}\) for the arithmetic sequence \(-17,-12,-7, \ldots\)