Sure, let’s solve this step by step:
1. We are given that the sequence is arithmetic, which means the difference between consecutive terms is constant.
2. We are provided with:
- [tex]\( a_{14} = -75 \)[/tex]
- [tex]\( a_{26} = -123 \)[/tex]
3. To find the common difference [tex]\( d \)[/tex], we use the formula for the nth term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
4. We need to find the value of [tex]\( d \)[/tex]. We know:
[tex]\[
a_{26} = a_{14} + 12d
\][/tex]
5. Plugging the given values into this equation:
[tex]\[
-123 = -75 + 12d
\][/tex]
6. Simplifying for [tex]\( d \)[/tex]:
[tex]\[
-123 + 75 = 12d
\][/tex]
[tex]\[
-48 = 12d
\][/tex]
[tex]\[
d = \frac{-48}{12}
\][/tex]
[tex]\[
d = -4
\][/tex]
7. Now we need to find the first term [tex]\( a_1 \)[/tex]. Using the value of [tex]\( a_{14} \)[/tex]:
[tex]\[
a_{14} = a_1 + 13d
\][/tex]
[tex]\[
-75 = a_1 + 13(-4)
\][/tex]
[tex]\[
-75 = a_1 - 52
\][/tex]
[tex]\[
a_1 = -75 + 52
\][/tex]
[tex]\[
a_1 = -23
\][/tex]
8. Now we check which recursive formula matches:
- [tex]\( a_n = a_{n-1} - 4 \)[/tex] has [tex]\( d = -4 \)[/tex] and from our calculation, [tex]\( a_1 = -23 \)[/tex].
Therefore, the correct recursive formula is:
[tex]\[ a_n = a_{n-1} - 4, \quad a_1 = -23 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{a_n = a_{n-1} - 4; \quad a_1 = -23} \][/tex]
