To determine which recursive formula could represent the given sequence, we need to check each of the provided recursive formulas using the terms [tex]\( a_4=24, a_3=8408, a_2=664, \)[/tex] and [tex]\( a_1=88 \)[/tex].
1. First Option: [tex]\( a_n = (a_{n-2})^2 + a_{n-1} \)[/tex]
- We will verify if this formula holds for [tex]\( n=4 \)[/tex]:
[tex]\( a_4 = (a_2)^2 + a_3 \)[/tex]
[tex]\[
a_4 = 664^2 + 8408
\][/tex]
[tex]\[
a_4 = 441856 + 8408 = 450264 \neq 24
\][/tex]
Since [tex]\( 450264 \neq 24 \)[/tex], this formula does not represent the sequence.
2. Second Option: [tex]\( a_n = 3a_{n-1} + 16 \)[/tex]
- We will verify if this formula holds for [tex]\( n=4 \)[/tex]:
[tex]\( a_4 = 3a_3 + 16 \)[/tex]
[tex]\[
a_4 = 3 \times 8408 + 16
\][/tex]
[tex]\[
a_4 = 25224 + 16 = 25240 \neq 24
\][/tex]
Since [tex]\( 25240 \neq 24 \)[/tex], this formula does not represent the sequence.
3. Third Option: [tex]\( a_n = n \cdot a_{n-1} - 8 \)[/tex]
- We will verify if this formula holds for [tex]\( n=4 \)[/tex]:
[tex]\( a_4 = 4 \cdot a_3 - 8 \)[/tex]
[tex]\[
a_4 = 4 \times 8408 - 8
\][/tex]
[tex]\[
a_4 = 33632 - 8 = 33624 \neq 24
\][/tex]
Since [tex]\( 33624 \neq 24 \)[/tex], this formula does not represent the sequence.
4. Fourth Option: [tex]\( a_n = 2a_{n-2} + 7a_{n-1} \)[/tex]
- We will verify if this formula holds for [tex]\( n=4 \)[/tex]:
[tex]\( a_4 = 2a_2 + 7a_3 \)[/tex]
[tex]\[
a_4 = 2 \times 664 + 7 \times 8408
\][/tex]
[tex]\[
a_4 = 1328 + 58856 = 60184 \neq 24
\][/tex]
Since [tex]\( 60184 \neq 24 \)[/tex], this formula does not represent the sequence.
Since none of the given formulas match the sequence's terms, we can conclude that none of the given recursive formulas accurately represent the given sequence. Therefore, the correct answer is that none of the provided options match the sequence.
