Certainly! Let's write the recursive sequence that corresponds to the given explicit formula:
[tex]\[ a_n = -7 + 10(n-1) \][/tex]
### Step-by-Step Solution:
1. Identify the first term:
The explicit formula is provided, which helps us find the first term when [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = -7 + 10 \times (1 - 1) \][/tex]
2. Simplify the expression for [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = -7 + 10 \times 0 \][/tex]
[tex]\[ a_1 = -7 \][/tex]
So, the first term of the sequence is:
[tex]\[ a_1 = -7 \][/tex]
3. Determine the recursive relationship:
From the explicit formula, we can see that each term increases by a fixed amount (10) compared to the previous term. To represent this in a recursive formula, we express [tex]\( a_n \)[/tex] in terms of [tex]\( a_{n-1} \)[/tex]:
Since we know:
[tex]\[ a_n = -7 + 10(n-1) \][/tex]
and for [tex]\( n-1 \)[/tex]:
[tex]\[ a_{n-1} = -7 + 10((n-1)-1) = -7 + 10(n-2) \][/tex]
Subtracting these two sequences:
[tex]\[ a_n - a_{n-1} = [-7 + 10(n-1)] - [-7 + 10(n-2)] \][/tex]
[tex]\[ a_n - a_{n-1} = 10(n-1) - 10(n-2) \][/tex]
[tex]\[ a_n - a_{n-1} = 10 \][/tex]
Therefore, each term [tex]\( a_n \)[/tex] is 10 more than the previous term [tex]\( a_{n-1} \)[/tex].
Now, we write the recursive formula:
[tex]\[ a_n = a_{n-1} + 10 \][/tex]
### Final Recursive Sequence:
Given the sequence's first term and the recursive relationship, the recursive sequence can be written as:
[tex]\[
a_1 = -7
\][/tex]
[tex]\[
a_n = a_{n-1} + 10 \text{ for } n > 1
\][/tex]
Thus, we have successfully converted the explicit formula into a recursive sequence.
