Answer:
[tex]a_n=a_{n-1} \times 1.05[/tex]
Step-by-step explanation:
In a recursive formula, each term of the sequence is found by using the previous term. In this case, the sequence is geometric because we are multiplying each term by the same constant, called a common ratio.
To find this common ratio, we find the percent of change between terms. We do this by using the formula
% change = (amount of change)/(original amount)
The amount of change between the first two terms is 105-100 = 5. This makes the percent change
5/100 = 0.05
The amount of change between the second two terms is 110.25-105 = 5.25. This makes the percent change
5.25/105 = 0.05
However, the common ratio in the sequence is not 0.05. This is because multiplying by this decimal would make the terms decrease instead of increasing. In this case, we are taking 100% of the number and adding an extra 5% (0.05 = 5%); this makes our common ratio 105% = 1.05.
This gives us the recursive formula
[tex]a_n=a_{n-1} \times 1.05[/tex].
The question illustrates geometric function, because it uses a common ratio.
The recursive function is: [tex]T_n = T_{n-1} \times 1.05[/tex]
We have:
[tex]T_1 = 100[/tex]
[tex]T_2 = 105[/tex]
[tex]T_3 = 110.25[/tex]
Rewrite the second and third term as:
[tex]T_2 = 100 \times 1.05[/tex]
[tex]T_3 = 105 \times 1.05[/tex]
Substitute [tex]T_2 = 105[/tex] in [tex]T_3 = 105 \times 1.05[/tex]
[tex]T_3 = T_2 \times 1.05[/tex]
Express 2 as 3 -1
[tex]T_3 = T_{3-1} \times 1.05[/tex]
Substitute n for 3, for the nth term
[tex]T_n = T_{n-1} \times 1.05[/tex]
Hence, the recursive function is:
[tex]T_n = T_{n-1} \times 1.05[/tex]
Read more about recursive function at:
https://brainly.com/question/13657607
