12.1 Sequences

Before you get started, take this readiness quiz.

Evaluate $2 n + 3$ for the integers 1, 2, 3, and 4.
If you missed this problem, review the reference.

Evaluate $\left(−1\right)^{n}$ for the integers 1, 2, 3, and 4.
If you missed this problem, review the reference.

If $f \left(n\right) = n^{2} + 2 ,$ find $f \left(1\right) + f \left(2\right) + f \left(3\right) .$
If you missed this problem, review the reference.

Write the first five terms of the sequence whose general term is $a_{n} = 3 n - 4 .$

Write the first five terms of the sequence whose general term is $a_{n} = 2 n - 5 .$

Write the first five terms of the sequence whose general term is $a_{n} = 3^{n} + 4 .$

Write the first five terms of the sequence whose general term is $a_{n} = 2^{n} - 5 .$

Write the first five terms of the sequence whose general term is $a_{n} = \left(−1\right)^{n} n^{2} .$

Write the first five terms of the sequence whose general term is $a_{n} = \left(−1\right)^{n + 1} n^{3} .$

Find a general term for the sequence whose first five terms are shown.

$3 , 6 , 9 , 12 , 15 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$5 , 10 , 15 , 20 , 25 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$−3 , 9 , −27 , 81 , −243 , \ldots$

Find a general term for the sequence whose first five terms are shown

$1 , −4 , 9 , −16 , 25 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{2} , \frac{1}{4} , \frac{1}{8} , \frac{1}{16} , \frac{1}{32} , \ldots$

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{1} , \frac{1}{4} , \frac{1}{9} , \frac{1}{16} , \frac{1}{25} , \ldots$

Write the first five terms of the sequence whose general term is $a_{n} = \frac{2}{n !} .$

Write the first five terms of the sequence whose general term is $a_{n} = \frac{3}{n !} .$

Write the first five terms of the sequence whose general term is $a_{n} = \frac{\left(n - 1\right) !}{\left(n + 1\right) !} .$

Write the first five terms of the sequence whose general term is $a_{n} = \frac{n !}{\left(n + 1\right) !} .$

Expand the partial sum and find its value: $\sum_{i = 1}^{5} 2 i .$

Expand the partial sum and find its value: $\sum_{i = 1}^{5} 3 i .$

Expand the partial sum and find its value: $\sum_{i = 1}^{5} 4 i .$

Expand the partial sum and find its value: $\sum_{k = 0}^{3} \frac{1}{k !} .$

Expand the partial sum and find its value: $\sum_{k = 0}^{3} \frac{2}{k !} .$

Expand the partial sum and find its value: $\sum_{k = 0}^{3} \frac{3}{k !} .$

Write the sum using summation notation: $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} .$

Write the sum using summation notation: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} .$

Write the sum using summation notation: $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} .$

Write each sum using summation notation: $1 - 4 + 9 - 16 + 25 .$

Write each sum using summation notation: $−2 + 4 - 6 + 8 - 10 .$

Write the first five terms of the sequence whose general term is given.

$a_{n} = 2 n - 7$

Write the first five terms of the sequence whose general term is given.

$a_{n} = 3 n + 1$

Write the first five terms of the sequence whose general term is given.

$a_{n} = 2^{n} + 3$

Write the first five terms of the sequence whose general term is given.

$a_{n} = 3^{n} - 2 n$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{2^{n}}{n^{2}}$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{4 n - 2}{2^{n}}$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \left(−1\right)^{n} \cdot 2 n$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \left(−1\right)^{n + 1} n^{2}$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{\left(−1\right)^{n + 1}}{n^{2}}$

Find a general term for the sequence whose first five terms are shown.

$8 , 16 , 24 , 32 , 40 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$6 , 7 , 8 , 9 , 10 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$e^{3} , e^{4} , e^{5} , e^{6} , e^{7} , \ldots$

Find a general term for the sequence whose first five terms are shown.

$−5 , 10 , −15 , 20 , −25 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$−1 , 8 , −27 , 64 , −125 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$−2 , 4 , −6 , 8 , −10 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{4} , \frac{1}{16} , \frac{1}{64} , \frac{1}{256} , \frac{1}{1,024} , \ldots$

Find a general term for the sequence whose first five terms are shown.

$- \frac{1}{2} , - \frac{2}{3} , - \frac{3}{4} , - \frac{4}{5} , - \frac{5}{6} , \ldots$

Find a general term for the sequence whose first five terms are shown.

$- \frac{5}{2} , - \frac{5}{4} , - \frac{5}{8} , - \frac{5}{16} , - \frac{5}{32} , \ldots$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{4}{n !}$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = 3 n !$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = \left(2 n\right) !$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{\left(n - 1\right) !}{\left(n\right) !}$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{n !}{n^{\text{2}}}$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{\left(n + 1\right) !}{n^{2}}$

Expand the partial sum and find its value.

$\sum_{i = 1}^{5} i^{2}$

Expand the partial sum and find its value.

$\sum_{i = 1}^{6} \left(\right. 2 i + 3 \left.\right)$

Expand the partial sum and find its value.

$\sum_{i = 1}^{4} 2^{i}$

Expand the partial sum and find its value.

$\sum_{k = 0}^{3} \frac{4}{k !}$

Expand the partial sum and find its value.

$\sum_{k = 1}^{5} k \left(k + 1\right)$

Expand the partial sum and find its value.

$\sum_{n = 1}^{5} \frac{n}{n + 1}$

Write each sum using summation notation.

$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243}$

Write each sum using summation notation.

$1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \frac{1}{125}$

Write each sum using summation notation.

$2 + 1 + \frac{2}{3} + \frac{1}{2} + \frac{2}{5}$

Write each sum using summation notation.

$3 - 6 + 9 - 12 + 15$

Write each sum using summation notation.

$−2 + 4 - 6 + 8 - 10 + \ldots + 20$

Write each sum using summation notation.

$14 + 16 + 18 + 20 + 22 + 24 + 26$

In your own words, explain how to write the terms of a sequence when you know the formula. Show an example to illustrate your explanation.

In your own words, explain what is meant by $n !$ Show some examples to illustrate your explanation.