Answer:
an = 1/2n² -1/2n -2
Explanation:
You want a formula for the n-th term of the quadratic sequence -2, -1, 1, 4.
Linear equations
Using x = 1, 2, 3 and y = -2, -1, 1 we can write equations for the coefficients a, b, c of the quadratic expression ax² +bx +c.
a(1²) +b(1) +c = -2
a(2²) +b(2) +c = -1
a(3²) +b(3) +c = 1
Solution
Subtracting the first equation from the other two, we get ...
3a +b = 1
8a +2b = 3
Subtracting twice the first from the second of these, we get ...
(8a +2b) -2(3a +b) = (3) -2(1)
2a = 1
a = 1/2
Then b can be found from ...
b = 1 -3a = 1 - 3/2 = -1/2
And c can be found from ...
-2 -b -a = c = -2 -(-1/2) -(1/2) = -2
The formula is ...
an = 1/2n² -1/2n -2
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Additional comment
For systems of equations, the solver used in the second attachment works well. It tells us (a, b, c) = (1/2, -1/2, -2) as we found above.
Alternate solution
The first differences of the given terms are ...
1, 2, 3
And their differences are ...
1, 1
The coefficient of n² is half this second-difference value: 1/2.
If you subtract 1/2n² from the given terms, you get ...
-5/2, -3, -7/2, -4
This arithmetic sequence has the formula ...
a1 +d(n -1)
-5/2 +(-1/2)(n -1) = -1/2n -2
Adding this to the term we subtracted gives ...
an = 1/2n² -1/2n -2 . . . . . as above