The correct option will be: A. [tex] a_{n}=6n-13 [/tex]
Explanation
Given that, [tex] a_{10}=47, a_{11}=53, a_{12}=59, a_{13}=65 [/tex]
So, the common difference in the arithmetic sequence is: [tex] d=a_{11} -a_{10}= 53-47= 6 [/tex]
Formula for finding n-th term in arithmetic sequence is: [tex] a_{n}=a_{1} +(n-1)d [/tex] , where [tex] a_{1} [/tex] is the first term of the sequence.
According to this formula...
[tex] a_{10}=a_{1}+(10-1)*6\\\\ 47=a_{1}+(9)*6 \\\\ 47= a_{1}+54\\\\ a_{1}= 47-54=-7 [/tex]
So, the first term of the sequence is -7
Again using the formula of n-th term, we will get...
[tex] a_{n}= a_{1}+(n-1)d\\\\ a_{n}=-7+(n-1)*6\\\\ a_{n}=-7+6n-6\\\\ a_{n}=6n-13 [/tex]
So, the formula can be used to find [tex] a_{n} [/tex] is: [tex] a_{n}= 6n-13 [/tex]
