Answer :
Answer: The sequence is an arithmetic sequence
The recursive formula is given as [tex]a_{n[/tex] = [tex]a_{n-1}[/tex] + 7 , where [tex]a_{1}[/tex] = 14
Step-by-step explanation:
[tex]\\[/tex]For a sequence to be geometric then there must exist a common ratio. Let r represent the common ratio. The formula for calculating common ratio implies:
[tex]\\[/tex]r = [tex]\frac{T_{2} }{T_{1} }[/tex] = [tex]\frac{T_{3} }{T_{2} }[/tex] … , that is r is calculated by dividing the second term by the first term or the third term divided by the second term and so on.
[tex]\\[/tex]To check if the sequence is geometric, let us find the common ratio.
[tex]\\[/tex][tex]T_{1}[/tex] = 14
[tex]\\[/tex][tex]T_{2}[/tex] = 21
[tex]\\[/tex][tex]T_{3}[/tex] = 28
[tex]\\[/tex][tex]T_{4}[/tex] = 35
[tex]\\[/tex]So, [tex]\frac{T_{2} }{T_{1} }[/tex] = [tex]\frac{21}{14}[/tex] = [tex]\frac{3}{2}[/tex]
[tex]\\[/tex] [tex]\frac{T_{3} }{T_{2} }[/tex]= [tex]\frac{28}{21}[/tex] = [tex]\frac{4}{3}[/tex]
[tex]\\[/tex] [tex]\frac{T_{4} }{T_{3} }[/tex] = [tex]\frac{35}{28}[/tex] = [tex]\frac{5}{4}[/tex]
[tex]\\[/tex]Considering the result, it is clear that it is not a geometric sequence since the ratios are not the same
[tex]\\[/tex]Arithmetic Sequence , for a sequence to be arithmetic then there must be an existence of a common difference.That is
[tex]\\[/tex][tex]T_{2}[/tex] – [tex]T_{1}[/tex]= [tex]T_{3}[/tex] – [tex]T_{2}[/tex] = [tex]T_{4}[/tex]– [tex]T_{}[/tex]
[tex]\\[/tex]Let us check if the sequence given follow this rule
[tex]\\[/tex][tex]T_{2}[/tex] – [tex]T_{1}[/tex] = 21 -14 = 7
[tex]\\[/tex][tex]T_{3}[/tex] – [tex]T_{2}[/tex] = 28 – 21 = 7
[tex]\\[/tex][tex]T_{4}[/tex] – [tex]T_{}[/tex] = 35 – 28 = 7
[tex]\\[/tex]Therefore the sequence is an arithmetic sequence.
[tex]\\[/tex]To find the recursive formula for the sequence
[tex]\\[/tex] [tex]a_{1}[/tex] = 14
[tex]\\[/tex][tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] +d
[tex]\\[/tex][tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] +7