To identify the 42nd term of an arithmetic sequence, given that the first term [tex]\(a_1 = -12\)[/tex] and the 27th term [tex]\(a_{27} = 66\)[/tex], follow these steps:
1. Identify the formula for the nth term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
2. Use the information for the 27th term to determine the common difference [tex]\(d\)[/tex]:
- Given [tex]\(a_{27} = 66\)[/tex],
- Substitute into the formula: [tex]\( 66 = -12 + (27-1) \cdot d \)[/tex],
- Simplify the equation: [tex]\( 66 = -12 + 26 \cdot d \)[/tex],
- Solve for [tex]\(d\)[/tex]:
[tex]\[
66 + 12 = 26 \cdot d \implies 78 = 26 \cdot d \implies d = \frac{78}{26} = 3.
\][/tex]
3. Calculate the 42nd term using the found value of the common difference:
- We need to find [tex]\(a_{42}\)[/tex],
- Use the nth term formula: [tex]\( a_{42} = a_1 + (42-1) \cdot d \)[/tex],
- Substitute the known values:
[tex]\[
a_{42} = -12 + (42-1) \cdot 3,
\][/tex]
- Simplify the equation:
[tex]\[
a_{42} = -12 + 41 \cdot 3 ,
\][/tex]
- Calculate the value:
[tex]\[
a_{42} = -12 + 123 = 111.
\][/tex]
So, the 42nd term of the arithmetic sequence is 111. Therefore, the correct answer is:
111
