Answer:
f(n)=n²-3
Explanation:
In the sequence:
[tex]-2,1,6,13,22,...[/tex]
First, we find the difference between the terms.
[tex]\begin{gathered} 1-(-2)=3 \\ 6-1=5 \\ 13-6=7 \\ 22-13=9 \end{gathered}[/tex]
It is observed that the difference between successive terms is the addition of consecutive odd numbers.
This is an example of a quadratic sequence.
The general form of a quadratic sequence is:
[tex]\begin{gathered} f(n)=an^2+bn+c \\ f(1)=-2 \\ \implies a+b+c=-2 \\ f(2)=1 \\ \implies4a+2b+c=1 \\ f(3)=6 \\ \implies9a+3b+c=6 \end{gathered}[/tex]
If we solve the system of equations:
[tex]\begin{gathered} a+b+c=-2 \\ 4a+2b+c=1 \\ 9a+3b+c=6 \\ a=1,b=0,c=-3 \end{gathered}[/tex]
The explicit expression for this sequence is:
[tex]f(n)=n^2-3[/tex]
