Looking at the graph, the three consecutive elements of the sequence are:
[tex]2,7,24.5[/tex]
(a)
The common ratio r is the division between two consecutive elements, with the n-th element as the denominator and the (n+1)-th element as the numerator. From the list of elements, the common ratio is:
[tex]r=\frac{7}{2}=\frac{24.5}{7}=3.5[/tex]
(b)
The initial value of the sequence is 2, and the common ratio is 3.5, The general recursive formula can be expressed as:
[tex]\begin{gathered} a_1=b \\ a_{n+1}=a_n\cdot r,\text{ for }n\ge1 \end{gathered}[/tex]
Now, for our problem, we identify b = 2 and r = 3.5. The recursive formula is:
[tex]\begin{gathered} a_1=2 \\ a_{n+1}=3.5\cdot a_n,\text{ for }n\ge1 \end{gathered}[/tex]
( )
The explicit formula of the sequence is:
[tex]a_n=a_1\cdot r^{n-1},\text{ for }n\ge1[/tex]
Using the values of a₁ and r:
[tex]a_n=2\cdot3.5^{n-1},\text{ for }n\ge1[/tex]
(d)
The eighth term of the sequence can be calculated if we set n = 8 in the previous formula:
[tex]\begin{gathered} a_8=2\cdot3.5^{8-1}=2\cdot3.5^7 \\ \therefore a_8=12867.9 \end{gathered}[/tex]
