Answer:
[tex]g(n) = 6n -25[/tex]
Step-by-step explanation:
Given:
[tex]g(1) = -19[/tex]
[tex]g(n) = g(n - 1) + 6[/tex]
Required
The explicit formula
Let n = 2; So, we have:
[tex]g(n) = g(n - 1) + 6[/tex]
[tex]g(2) = g(2 - 1) + 6[/tex]
[tex]g(2) = g(1) + 6[/tex]
[tex]g(2) = -19 + 6[/tex]
[tex]g(2) = -13[/tex]
So, we have:
[tex]g(1) = -19[/tex] ----- First term
[tex]g(2) = -13[/tex]
Calculate common difference (d)
[tex]d = g(2) - g(1)[/tex]
[tex]d = -13 --19[/tex]
[tex]d = 6[/tex]
The explicit function is then calculated as:
[tex]g(n) = a + (n - 1)d[/tex]
Where
[tex]a = -19[/tex] --- First term
So:
[tex]g(n) = a + (n - 1)d[/tex]
[tex]g(n) = -19 + (n - 1)*6[/tex]
Open bracket
[tex]g(n) = -19 + 6n - 6[/tex]
Collect like terms
[tex]g(n) = 6n - 6-19[/tex]
[tex]g(n) = 6n -25[/tex]
