To find the missing number in the arithmetic sequence given by [tex]\(16, \ldots, 52\)[/tex], let's follow a detailed, step-by-step solution.
1. Identify the positions and terms:
- The first term [tex]\(a_1\)[/tex] is [tex]\(16\)[/tex].
- The term you're looking for [tex]\(a_2\)[/tex] is the missing number.
- The third term [tex]\(a_3\)[/tex] is [tex]\(52\)[/tex].
2. Use the formula for the nth term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Set up the equation for the third term ([tex]\(a_3\)[/tex]):
[tex]\[ a_3 = a_1 + 2d \][/tex]
Substitute the known values:
[tex]\[ 52 = 16 + 2d \][/tex]
4. Solve for [tex]\(d\)[/tex]:
[tex]\[ 52 - 16 = 2d \][/tex]
[tex]\[ 36 = 2d \][/tex]
[tex]\[ d = \frac{36}{2} \][/tex]
[tex]\[ d = 18 \][/tex]
5. Use the common difference [tex]\(d\)[/tex] to find the second term ([tex]\(a_2\)[/tex]):
[tex]\[ a_2 = a_1 + (2-1)d \][/tex]
Substitute the known values:
[tex]\[ a_2 = 16 + 1 \cdot 18 \][/tex]
[tex]\[ a_2 = 16 + 18 \][/tex]
[tex]\[ a_2 = 34 \][/tex]
Therefore, the missing number in the arithmetic sequence [tex]\(16, \ldots, 52\)[/tex] is:
[tex]\[ \boxed{34} \][/tex]
