Answer:
[tex]f(n) = 5 + f(n - 1)[/tex]
Step-by-step explanation:
Given
[tex]f(n) = 5n - 2[/tex]
Required
Write a recursive expression
[tex]f(n) = 5n - 2[/tex]
When n = 1
[tex]f(1) = 5 * 1 - 2 = 5 - 2 = 3[/tex]
[tex]f(1) = 3[/tex]
When n = 2
[tex]f(2) = 5 * 2 - 2 = 10 - 2 = 8[/tex]
[tex]f(2) = 8[/tex] can be rewritten as
[tex]f(2) = 5 + 3[/tex]
Substitute 3 for f(1)
[tex]f(2) = 5 + f(1)[/tex]
Express 1 as 2 - 1
[tex]f(2) = 5 + f(2-1)[/tex] ----- (1)
When n = 3
[tex]f(3) = 5 * 3 - 2 = 15 - 2 = 13[/tex]
[tex]f(3) = 13[/tex] can be rewritten as
[tex]f(3) = 5 + 8[/tex]
Substitute 8 for f(2)
[tex]f(3) = 5 + f(2)[/tex]
Express 2 as 3 - 1
[tex]f(3) = 5 + f(3 - 1)[/tex] ------ (2)
Write out (1) and (2)
[tex]f(2) = 5 + f(2-1)[/tex]
[tex]f(3) = 5 + f(3 - 1)[/tex]
Replace 2 and 3 with n in both cases; This gives
[tex]f(n) = 5 + f(n - 1)[/tex]
Hence;
The recursive is [tex]f(n) = 5 + f(n - 1)[/tex]
