Please help me i need this for today

The nth term is

n is any positive integer (1,2,3,...)

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Work Shown:

The jump from -5 to -2 is +3

The jump from -2 to 3 is +5

The jump from 3 to 10 is +7

The jump from 10 to 19 is +9

Focusing on the increases we see: 3, 5, 7, 9

This is an arithmetic sequence of its own. It steadily increases by 2 each time. Because of this, we can say that the original sequence is a quadratic one.

It is of the form an^2 + bn + c

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Plug in n = 1 to get

an^2 + bn + c = a(1)^2 + b(1) + c = a+b+c

This is set equal to -5 as this is the first term; meaning we have a+b+c = -5

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Now plug in n = 2

an^2 + bn + c = a(2)^2 + b(2) + c = 4a+2b+c

Set this equal to -2, as this is the second term, getting 4a+2b+c = -2

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Plug in n = 3 and follow the same basic steps as before

an^2 + bn + c = a(3)^2 + b(3) + c = 9a+3b+c

We get 9a+3b+c = 3 due to 3 being the third term

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The system of equations we end up with are

Solve this system however you like. One option is to use a matrix (either inverse or RREF), or you could use elimination. Graphing is not really feasible here. Whichever method you pick, you should end up with the solution (a,b,c) = (1,0,-6)

Meaning, a = 1, b = 0, c = -6

Therefore, an^2 + bn + c would update to 1n^2+0n+(-6) which is the same as n^2 - 6 which is the nth term we're after

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As a check, plug in n = 1 and see what happens. We should get -5 as a result

n^2 - 6 = 1^2 - 6 = -5

Checks out. Let's try n = 2

n^2 - 6 = 2^2 - 6 = -2

That works as well. I'll let you check the others.

the height of the surface increases from the center out to the sides of the road.

step-by-step explanation:

as we move out we move away from the x-axis.

and as mentioned in the problem statement that height of a road increases when we move away from the center x.

hence proved that 3rd option

the height of the surface increases from the center out to the sides of the road.

is the correct option.