Write a rule to find the nth term for an arithmetic sequence given the following:a3 = 14a12 = 59Recursive rule: ? Explicit rule: ?Write an explicit and recursive rule for a sequence given the following: a4 = 2r = 1/3Recursive rule: ? Explicit rule: ?

ANSWERSee explanationEXPLANATIONQuestion 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=9dThis implies thatd=5a=14-2(5)=4The 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 2The geometric sequence has the fourth term to be 2 and the common ratio to be r=⅓This 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]