Answer:
Option B - [tex]a_n=2a_{n-1}+3 ; a_1=5[/tex]
Step-by-step explanation:
Given : Sequence 5, 13, 29, 61, 125, ...
To find : Which recursive formula describes the sequence below?
Solution :
To find which function gives recursive formula substitute the value and check which satisfy the given sequence.
A. [tex]a_n=a_{n-1}+8 ; a_1=5[/tex]
Finding 2 term put n=2
[tex]a_2=a_{2-1}+8[/tex]
[tex]a_2=a_{1}+8[/tex]
[tex]a_{2}= 5+8[/tex]
[tex]a_{2}= 13[/tex]
Applicable as second term is 13.
Check for third term put n=3
[tex]a_3=a_{3-1}+8[/tex]
[tex]a_3=a_{2}+8[/tex]
[tex]a_{3}= 13+8[/tex]
[tex]a_{3}= 21[/tex]
Not applicable as third term is 29.
B. [tex]a_n=2a_{n-1}+3 ; a_1=5[/tex]
Finding 2 term put n=2
[tex]a_2=2a_{2-1}+3[/tex]
[tex]a_2=2a_{1}+3[/tex]
[tex]a_{2}= 2(5)+3[/tex]
[tex]a_{2}= 13[/tex]
Applicable as second term is 13.
Check for third term put n=3
[tex]a_3=2a_{3-1}+3[/tex]
[tex]a_3=2a_{2}+3[/tex]
[tex]a_{3}= 2(13)+3[/tex]
[tex]a_{3}= 26+3[/tex]
[tex]a_{3}= 29[/tex]
Applicable as third term is 29.
C. [tex]a_n=3a_{n-1}-2 ; a_1=5[/tex]
Finding 2 term put n=2
[tex]a_2=3a_{2-1}-2[/tex]
[tex]a_2=3a_{1}-2[/tex]
[tex]a_{2}= 3(5)-2[/tex]
[tex]a_{2}= 13[/tex]
Applicable as second term is 13.
Check for third term put n=3
[tex]a_3=3a_{3-1}-2[/tex]
[tex]a_3=3a_{2}-2[/tex]
[tex]a_{3}= 3(13)-2[/tex]
[tex]a_{3}= 37[/tex]
Not applicable as third term is 29.
D. [tex]a_n=a_{n-1}^2-12 ; a_1=5[/tex]
Finding 2 term put n=2
[tex]a_2=a_{2-1}^2-12[/tex]
[tex]a_2=a_{1}^2-12[/tex]
[tex]a_{2}= 5^2-12[/tex]
[tex]a_{2}= 13[/tex]
Applicable as second term is 13.
Check for third term put n=3
[tex]a_3=a_{3-1}^2-12[/tex]
[tex]a_3=a_{2}^2-12[/tex]
[tex]a_{3}= (13)^2-12[/tex]
[tex]a_{3}= 157[/tex]
Not applicable as third term is 29.
Therefore, Only Option B is correct as it form the recursive formula of given sequence.
