Answer:
[tex]a_{n}=45-3n[/tex]
Step-by-step explanation:
Method 1:
Arithmetic sequence is in the form
[tex]a_{n} =a_{1} +(n-1)d\\[/tex]
d is the common difference, can be found by:
[tex]d=a_{n}-a_{n-1}=-3[/tex]
Subtituting the [tex]a_{1}[/tex] and [tex]d[/tex]
You get:
[tex]a_{n}=42+(-3)(n-1)=45-3n[/tex]
Method 2 (Mathematical induction):
Assume it is in form [tex]a_{n}=45-3n[/tex]
Base step: [tex]a_{1} =45-3(1)=42[/tex]
Inducive hypophesis: [tex]a_{n}=45-3n[/tex]
GIven: [tex]a_{n+1} =a_{n}-3[/tex]
[tex]a_{n+1}=45-3n-3=45-3(n+1)[/tex]
Proved by mathematical induction
[tex]a_{n}=45-3n[/tex]
