To identify the 42nd term of an arithmetic sequence where the first term [tex]\( a_1 = -12 \)[/tex] and the 27th term [tex]\( a_{27} = 66 \)[/tex], follow these steps:
1. Set up the formula for the nth term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/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.
2. Write the relationship for the 27th term:
[tex]\[
a_{27} = a_1 + 26d
\][/tex]
Plug in the given values:
[tex]\[
66 = -12 + 26d
\][/tex]
3. Solve for the common difference [tex]\( d \)[/tex]:
[tex]\[
66 + 12 = 26d
\][/tex]
[tex]\[
78 = 26d
\][/tex]
[tex]\[
d = \frac{78}{26} = 3
\][/tex]
4. Now, find the 42nd term using the nth term formula:
[tex]\[
a_{42} = a_1 + (42 - 1)d
\][/tex]
[tex]\[
a_{42} = -12 + 41 \cdot 3
\][/tex]
5. Calculate the value:
[tex]\[
a_{42} = -12 + 123
\][/tex]
[tex]\[
a_{42} = 111
\][/tex]
Therefore, the 42nd term of the arithmetic sequence is 111.
