Answer:
[tex]a_{n} = 27 - 3(n-1)[/tex]
Step-by-step explanation:
Arithmetic sequences concepts:
The general rule of an arithmetic sequence is the following:[tex]a_{n+1} = a_{n} + d[/tex]
In which d is the common diference between each term.
We can expand the general equation to find the nth term from the first, by the following equation:
[tex]a_{n} = a_{1} + (n-1)*d[/tex]
And also:
[tex]a_{n} = a_{m} + (n-m)*d[/tex]
In this question:
[tex]a_{4} = 18, a_{7} = 9[/tex]
Finding the common ratio:
[tex]a_{n} = a_{m} + (n-m)*d[/tex]
[tex]a_{7} = a_{4} + (7-4)*d[/tex]
[tex]9 = 18 + 3d[/tex]
[tex]3d = -9[/tex]
[tex]d = \frac{-9}{3}[/tex]
[tex]d = -3[/tex]
Finding the first term:
[tex]a_{4} = a_{1} + (4-1)*d[/tex]
[tex]18 = a_{1} + 3*(-3)[/tex]
[tex]a_{1} = 27[/tex]
General rule:
[tex]a_{n} = a_{1} + (n-1)*d[/tex]
[tex]a_{n} = 27 - 3(n-1)[/tex]
