quadratic sequence
seems to be
[tex]a_n=an^2+bn+c[/tex]
where [tex]a_1=-8[/tex], [tex]a_2=-8[/tex] and so on
we only need 3 points
(1,-8)
(2,-8)
(3,6)
subsitute and solve
(1,-8)
[tex]-8=a(1)^2+b(1)+c[/tex]
[tex]-8=a+b+c[/tex]
(2,-8)
[tex]-8=a(2)^2+b(2)+c[/tex]
[tex]-8=4a+2b+c[/tex]
(3,6)
[tex]6=a(3)^2+b(3)+c[/tex]
[tex]6=9a+3b+c[/tex]
so we have
-8=a+b+c
-8=4a+2b+c
6=9a+3b+c
add negative of the first equation to the last 2 equations to get
-8=a+b+c
0=3a+b
14=8a+2b
work with the last 2 equations
0=3a+b
14=8a+2b
multiply 1st equation by -2 and add to 2nd
0=-6a-2b
14=8a+2b +
14=2a+0
14=2a
divide by 2
7=a
sub back
0=3a+b
0=3(7)+b
0=21+b
-21=b
sub back
-8=a+b+c
-8=7-21+c
-8=-14+c
6=c
so a=7, b=-21, c=6
the nth term is [tex]a_n=7n^2-21n+6[/tex]
or in function form, [tex]f(n)=7n^2-21n+6[/tex]
