Answer:
[tex]n^2 -3n - 6[/tex]
Step-by-step explanation:
Looking at the image attached, you must first work out the difference between each term. This is +0, +2, +4, +6, +8, +10 (shown in orange).
Because you now have a linear sequence, the difference is now the same each time. This is +2 (shown in blue).
Because the original sequence is a quadratic, you have to halve the second difference (the +2). This means that the value of [tex]n^2[/tex] is 1, so 1[tex]n^2[/tex] or just [tex]n^2[/tex] .
If you write out the values of [tex]n^2[/tex], you can work out how much more you need to add or take away. For example, take the first three terms in the sequence.
n = 1, 2, 3
x = -8, -8, -6
[tex]n^2[/tex] = 1, 4, 9
Work out the difference between [tex]n^2[/tex] and x:
1 - -8 = 9
4 - -8 = 12
9 - -6 = 15
This gives you yet another linear sequence, 9, 12, 15.
Working out the formula for this, the difference is 3 each time, so it is 3n. The first value of n is 1, so 3n is 1. The difference is 6, so the formula for this linear sequence is 3n + 6.
Because [tex]n^2[/tex] is greater than x, you need to take 3n + 6 away from [tex]n^2[/tex].
This gives [tex]n^2[/tex] - (3n+6), so the final answer is:
[tex]n^2[/tex] - 3n - 6.
