Investigation: General Term of Arithmetic Sequences
Completion requirements
| Investigation |
General Term of Arithmetic Sequences
Think back to the Warm Up about alkanes. Let \(t_n \) represent the number of hydrogen atoms, \(n \) represent the number of carbon atoms, and \(d \) represent the common difference of \(2\). By analyzing these components, you can come up with a formula, or general term. The sequence is given.
Number of hydrogen atoms: \(4, 6, 8, 10\)
| Term |
Number of hydrogen atoms
|
\(d \)
(blue) |
\(n \)
(red) |
General Term, \(t_n \)
|
|
\(t_1 \)
|
\(4 \)
|
\(4 + 0({\color{blue}{2}}) \)
|
\(4 + ({\color{red}{1}} - 1)2 \) |
\(t_1 + (n - 1)d \)
|
|
\(t_2 \)
|
\(6 \)
|
\(4 + 1({\color{blue}{2}}) \) | \(4 + ({\color{red}{2}} - 1)2 \) | \(t_1 + (n - 1)d \) |
|
\(t_3 \)
|
\(8 \)
|
\(4 + 2({\color{blue}{2}}) \) | \(4 + ({\color{red}{3}} - 1)2 \) | \(t_1 + (n - 1)d \) |
|
\(t_4 \)
|
\(10 \)
|
\(4 + 3({\color{blue}{2}}) \) | \(4 + ({\color{red}{4}} - 1)2 \) | \(t_1 + (n - 1)d \) |
We can continue this pattern, of write the general term, \(t_1 + (n - 1)d \).