When we substitute [tex] n = 4 [/tex], we get...
[tex] a_{n-3}=a_1\\ \\ a_{n-2}=a_2\\ \\ a_{n-1}=a_3\\ \\ a_n=a_4 [/tex]
Comparing the given sequence: 24, 88, 664, 8408, we get
[tex] a_{n-3}=a_1=24\\ \\ a_{n-2}=a_2=88\\ \\ a_{n-1}=a_3=664\\ \\ a_n=a_4=8408 [/tex]
We need to substitute the above values in each option and check which is TRUE.
----
Option A
[tex] a_n=(a_{n-2})^2+a_{n-1}, a_1=2\\ \\ a_n=(88)^2+664=7744+664\\\\a_n=8408 (TRUE) [/tex]
Option B
[tex] a_n=3a_{n-1}+16\\ \\ a_n=3(664)+16\\\\a_n=2008 (FALSE) [/tex]
Option C
[tex] a_n=(n)a_{n-1}-8\\ \\
a_4=(4)a_3-8=4(664)-8\\\\a_4=2648 (FALSE) [/tex]
Option D
[tex] a_n=2a_{n-2}+7a_{n-1}\\ \\ a_n=2(88)+7(664)\\ \\ a_n=4824(FALSE) [/tex]
Conclusion:
Option A is the correct answer.
