Certainly! Let's work through the given sequence step-by-step to find the relevant values.
The sequence provided is:
[tex]\[ 12, 112, 212, 312, \ldots \][/tex]
1. Identify [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = 12 \][/tex]
The first term of the sequence is 12.
2. Determine the common difference [tex]\( d \)[/tex]:
To find the common difference, we subtract the first term from the second term:
[tex]\[ d = 112 - 12 = 100 \][/tex]
So, the common difference [tex]\( d \)[/tex] is 100.
3. Use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
The formula to find the [tex]\( n \)[/tex]-th term in an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
4. Calculate [tex]\( a_{13} \)[/tex]:
Here, [tex]\( n \)[/tex] is 13. We'll substitute [tex]\( a_1 = 12 \)[/tex], [tex]\( d = 100 \)[/tex], and [tex]\( n = 13 \)[/tex] into the formula:
[tex]\[
a_{13} = 12 + (13 - 1) \cdot 100
\][/tex]
Simplify the expression inside the parentheses first:
[tex]\[
a_{13} = 12 + 12 \cdot 100
\][/tex]
Now multiply:
[tex]\[
12 \cdot 100 = 1200
\][/tex]
Then add the initial term:
[tex]\[
a_{13} = 12 + 1200 = 1212
\][/tex]
Thus, the 13th term in the sequence is [tex]\( a_{13} = 1212 \)[/tex].
