ANSWER
See explanation
EXPLANATION
Question 1:
The third term of the arithmetic sequence is :
14=a+2d...(1)
The twelveth term is
59=a+11d...(2)
Subtract equation (1) from (2)
45=9d
This implies that
d=5
a=14-2(5)=4
The explicit rule is;
[tex]a_{n}=4 + 5(n - 1)[/tex]
[tex]a_{n}=4 + 5n -5[/tex]
[tex]a_{n} = 5n -1[/tex]
Recursive formula:
[tex]a_{n}=a_{n - 1} + 5[/tex]
Question 2
The geometric sequence has the fourth term to be 2 and the common ratio to be r=⅓
This implies that,
[tex]a {( \frac{1}{3} })^{3} = 2[/tex]
This implies that,
[tex] \frac{a}{27} = 2[/tex]
[tex]a = 54[/tex]
The explicit rule:
[tex]a_n=54 {( \frac{1}{3} })^{n - 1} [/tex]
The recursive rule is
[tex]a_n=( \frac{1}{3} )a_{n-1}[/tex]
where,
[tex]a_1 = 54[/tex]
