Alright, let's work through the problem step-by-step to find the third term ([tex]\(a_3\)[/tex]) in the given arithmetic sequence.
An arithmetic sequence is defined as a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. Given:
- [tex]\(a_1 = 10\)[/tex] (the first term)
- The common difference, [tex]\(d = 6\)[/tex]
The [tex]\(n\)[/tex]-th term of an arithmetic sequence can be found using the formula:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
To find the third term, [tex]\(a_3\)[/tex], we substitute [tex]\(n = 3\)[/tex] into the formula:
[tex]\[ a_3 = a_1 + (3-1) \cdot d \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ a_3 = 10 + 2 \cdot 6 \][/tex]
Next, perform the multiplication:
[tex]\[ a_3 = 10 + 12 \][/tex]
Finally, add the numbers together:
[tex]\[ a_3 = 22 \][/tex]
Therefore, the third term in the sequence is [tex]\(22\)[/tex].
