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 \((2x+3y)^{12}\)