Answer:
[tex]\sf a_n = 9n + 12 [/tex]
Step-by-step explanation:
To find the explicit rule for the sequence [tex]\sf a_n [/tex] defined by the recursive rule [tex]\sf a_n = a_{n-1} + 9 [/tex] with [tex]\sf a_1 = 21 [/tex], we'll use the formula for arithmetic sequences.
Given:
- Recursive rule: [tex]\sf a_n = a_{n-1} + 9 [/tex]
- Initial term: [tex]\sf a_1 = 21 [/tex]
We'll follow these steps to find the explicit rule:
Identify the Common Difference:
The common difference [tex]\sf d [/tex] is the constant amount by which consecutive terms in the sequence increase or decrease.
In this case, [tex]\sf d = 9 [/tex] because each term is obtained by adding 9 to the previous term.
Apply the Formula for Arithmetic Sequences:
The formula for the [tex]\sf n [/tex]-th term of an arithmetic sequence is:
[tex]\large\boxed{\boxed{\sf a_n = a_1 + (n - 1) \cdot d}} [/tex]
Substitute Known Values:
Substitute the given values [tex]\sf a_1 = 21 [/tex] and [tex]\sf d = 9 [/tex] into the formula.
Simplify:
Simplify the expression to get the explicit rule for the sequence.
Let's calculate it step by step:
[tex]\sf a_n = a_1 + (n - 1) \cdot d [/tex]
[tex]\sf a_n = 21 + (n - 1) \cdot 9 [/tex]
[tex]\sf a_n = 21 + 9n - 9 [/tex]
[tex]\sf a_n = 9n + 12 [/tex]
Therefore, the explicit rule for the sequence [tex]\sf a_n [/tex] is:
[tex]\large\boxed{\boxed{\sf \boxed{a_n = 9n + 12}}} [/tex]
