Answer:
The explicit rule for the sequence in the table above is:
[tex]f_n=10n+2[/tex]
∴ 2nd option i.e. [tex]f_n=10n+2[/tex] is the correct option.
Step-by-step explanation:
Given the table
Tickets 1 2 3 4
Total Cost 12 22 32 42
An arithmetic sequence has a constant difference 'd' and is defined by
[tex]f_n=f_1+\left(n-1\right)d[/tex]
where:
[tex]f_n[/tex] = nth term
[tex]f_1[/tex] = first term
[tex]n[/tex] = term position
[tex]d[/tex] = common difference
In our case, the Arithmetic sequence is
12, 22, 32, 42,...
computing the differences of all the adjacent terms
[tex]22 - 12 = 10, 32 - 22 = 10, 42 - 32 = 10[/tex]
The difference between all the adjacent terms is the same and equal to
[tex]d = 10[/tex]
Important Tip:
As the first term involves the cost of 1 ticket which is 12, thus
[tex]f_1=12[/tex]
Now using the nth term formula of the Arithmetic sequence
[tex]f_n=f_1+\left(n-1\right)d[/tex]
now substituting [tex]f_1=12[/tex] and [tex]d = 10[/tex] in the formula
[tex]\:fn=10\left(n-1\right)+12[/tex]
[tex]f_n=10n-10+12\:[/tex]
simplifying
[tex]f_n=10n+2[/tex]
Therefore, the explicit rule for the sequence in the table above is:
[tex]f_n=10n+2[/tex]
∴ 2nd option i.e. [tex]f_n=10n+2[/tex] is the correct option.
