An arithmetic sequence is defined by a constant difference between consecutive terms. The nth term of an arithmetic sequence can be expressed with the formula:

$f(n) = f(1) + (n - 1) \cdot d$

where:

• $f(1)$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Given:

• $f(1) = 4$
• $f(2) = 9$

To find $d$:

$d = f(2) - f(1) = 9 - 4 = 5$

Now, substituting back into the formula:

$f(n) = 4 + (n - 1) \cdot 5$

Simplifying gives:

$f(n) = 4 + 5n - 5$ $f(n) = 5n - 1$

So, the nth term of the arithmetic sequence is:

$f(n) = 5n - 1$

Questions:

1. What is the common difference in this sequence?
2. How can you find the 10th term using this formula?
3. What would happen if $f(1)$ were changed to a different value?
4. Can you give an example of another arithmetic sequence?
5. How does this definition apply to geometric sequences?

Tip:

To quickly identify the common difference, subtract the first term from the second term in an arithmetic sequence.