Answer:
[tex]f(n) = n^{2} - 6[/tex]
Step-by-step explanation:
The given sequence is -5, -2,3, 10, 19.
We can observe the following pattern:
-5+3=-2
-2+5=3
3+7=10
10+9=19
There is no constant difference among the terms.
So the sequence is quadratic.
Let the nth term be:
[tex]f(n) = a {n}^{2} + bn + c[/tex]
[tex]f(1) = a(1)^{2} + b(1) + c \\ a + b+ c = - 5 - - - (1)[/tex]
Also,
[tex]f(2) = a(2)^{2} + b(2) + c \\ [/tex]
[tex]4a + 2b + c = - 2[/tex]
and
[tex]f(3) = a(3)^{2} + b(3) + c \\ 9a + 3b + c = - 3 - - - (3)[/tex]
We solve the three equations simultaneously to get:
a=1,b=0, c=-6.
Therefore the nth term is
[tex]f(n) = (n)^{2} + 0 \times n - 6 \\ f(n) = (n)^{2} - 6[/tex]
