Find the first \(4\) terms of the sequence whose general term is \(a_n=2n-1\)

Find the first \(4\) terms of the sequence whose general term is \(a_n=\displaystyle\frac{1}{n+1}\)

Find the \(5^{\text{th}}\) and \(6^{\text{th}}\) terms of the sequence whose general term is \(a_n=\displaystyle\frac{(-1)^n}{n^2}\)

Find the first \(4\) terms of the sequence given recursively by \(a_1=4\) and \(a_n=5a_{n-1}\)

Find the general term for \(2, \displaystyle\frac{3}{8}, \displaystyle\frac{4}{27}, \displaystyle\frac{5}{64}, \ldots\)

Expand and simplify \(\displaystyle\sum_{i=1}^{5} \left(i^2-1\right)\)

Expand and simplify \(\displaystyle\sum_{i=3}^{6}(-2)^i\)

Expand \(\displaystyle\sum_{i=2}^{5}\left(x^i-3\right)\)

Write with summation notation \(1+3+5+7+9\)

Write with summation notation \(3+12+27+48\)

Write with summation notation \(\displaystyle\frac{x+3}{x^3}+ \displaystyle\frac{x+4}{x^4}+ \displaystyle\frac{x+5}{x^5}+ \displaystyle\frac{x+6}{x^6}\)

Mini Lecture
Expand and simplify.

1. \(\displaystyle\sum_{i=1}^{4}(2t+4)\)

2. \(\displaystyle\sum_{i=3}^{6}(-2)^i\)

3. \(\displaystyle\sum_{i=3}^{6}(x+i)^i\)

Write with summation notation.

4. \(\displaystyle\frac{3}{4}+\displaystyle\frac{4}{5}+\displaystyle\frac{5}{6}+\displaystyle\frac{6}{7}+\displaystyle\frac{7}{8}\)

5. \(4+8+16+32+64\)

Give the common difference \(d\) for the arithmetic sequence \(4, 10, 16, 22, \ldots\)

Find the common difference for \(100, 93, 86, 79, \ldots\)

Find the common difference for \(\displaystyle\frac{1}{2}, 1, \displaystyle\frac{3}{2}, 2, \ldots\)

Find the general term for \(7, 19, 13, 16, \ldots\)

Find the sum of the first \(10\) terms of \(2, 10, 18, 26, \ldots\)

Mini Lecture
Is the sequence arithmetic?

1. \(50, 45, 40, \ldots\)

2. \(1, 4, 9, 16, \ldots\)

3. If \(a_1=3\) and \(d=4\), find \(a_n\) and \(a_{24}\)

4. If \(a_6=17\) and \(a_{12}=29\), find \(a_1\), \(d\), and \(a_{30}\)

5. Find \(S_{100}\) for \(5, 9, 13, 17, \ldots\)

Find the common ratio for \(\displaystyle\frac{1}{2}, \displaystyle\frac{1}{4}, \displaystyle\frac{1}{8}, \displaystyle\frac{1}{16}, \ldots\)

Find the common ratio for \(\sqrt{3}, 3, 3\sqrt{3}, 9, \ldots\)

Find the general term for \(5, 10, 20, \ldots\)

Find the tenth term of the sequence \(3, \displaystyle\frac{3}{2}, \displaystyle\frac{3}{4}, \displaystyle\frac{3}{8}, \ldots\)

Find the sum of the first \(10\) terms of \(5, 15, 45, 135, \ldots\)

Find the sum of the infinite series \(\displaystyle\frac{1}{5}+ \displaystyle\frac{1}{10}+ \displaystyle\frac{1}{20}+ \displaystyle\frac{1}{40}+ \ldots\)

Mini Lecture
Is the sequence geometric?

1. \(1, 5, 25, 125, \ldots\)

2. \(\displaystyle\frac{1}{2}, \displaystyle\frac{1}{6}, \displaystyle\frac{1}{18}, \displaystyle\frac{1}{54}, \ldots\)

3. If \(a_1=4\) and \(r=3\), find \(a_n, a_{20},\) and \(S_{20}\)

4. Find \(a_{10}\) and \(S_{10}\) for \(\sqrt{2}, 2, 2\sqrt{2}, \ldots\) \(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{4}+\displaystyle\frac{1}{8}+\ldots=\)?

5. Show that \(0.444\ldots=\displaystyle\frac{1}{9}\)

Expand \((x-2)^3\)

Expand \((3x+2y)^4\)

Find the first three terms in the expansion of \((x+5)^9\)

Find the fifth term in the expansion of \((2x+3y)^{12}\)

Use Venn diagrams to check the following expression. \[A\cap(B\cup C)=(A\cap B)\cup (A\cap C)\]

Let \(A\) and \(B\) be two intersecting sets, neither of which is a subset of the other. Use a Venn diagram to illustrate the set \[\{x\mid x\in A\text{ and }x\notin B\}\]

How many different ways are there to arrange the letters in the word CHEMISTRY?

In how many ways can \(8\) people fill \(5\) chairs at a table?

How many three-person committees can be formed from people in the set \(\{\text{Pat, Diane, Tim, JoAnn}\}\)?

A jar contains \(4\) coins: a penny, a nickel, a dime, and a quarter. If \(2\) coins are selected, how many different amounts of money are possible?