# I am needing help with understanding this pre-calc question in my math. I tried looking over the provided examples with the unit, but i'm stuck. the question is: The terms 5x + 2, 7x - 4 and 10x + 6 are consecuative terms of an arithmetic sequence. Determain the value of x and state the three terms.

The consecutive terms of an arithmetic sequence are given as 5x+2,7x-4, and 10x+6.

Since this is an arithmetic sequence, the terms are a constant distance apart.

Thus (10x+6)-(7x-4)=(7x-4)-(5x+2)

3x+10=2x-6

x=-16

If x=-16, the terms are -78,-116,-154.

You could use the definition:

a(n)=5x+2

a(n+1)=7x-4

a(n+2)=10x+6

a(n+1)=an+d

7x-4=5x+2+d

d=2x-6

So a(n+2)=a(n+1)+d

10x+6=7x-4+2x-6

10x=6=9x-10

x=-16 as before.

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x=-16 so the terms are -78,-116,-154

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We can use the formula for a term of an arithmetic sequence:
`a_n = a_1 + (n - 1)d`

The ` 10x + 6 ` is the third term which is `a_3` , the `5x + 2` is the first term` a_1` .
Since `10x + 6` is `a_3` , `n = 3` . Applying the values that we have on the formula above, we will have:

`10x + 6 = 5x + 2 + (3 - 1)d`

`10x + 6 = 5x + 2 + 2d`

Putting all the terms with variables on one side, and the constant term on left side.
Note that moving a term will change the sign of that term.
`10x - 5x -2d = 2 - 6`
`5x - 2d = - 4 `     (first equation)

The `7x - 4` is the second term which is` a_2` , the `5x + 2` is the first term` a_1` .
Since `7x - 4 ` is `a_2` , `n = 2` .

Applying the values that we have on the formula above, we will have:

`7x - 4 = 5x + 2 + (2 - 1)d`
`7x - 4 = 5x + 2 + d`
`7x - 5x - d = 2 + 4`

`2x - d = 2`   (second equation)

Apply the Elimination Method.
Multiply the second equation by` -2` .

`-2(2x - d) = -2(2)`
`-4x + 2d = 4`

`-4x + 2d = -4`
(+) `5x - 2d = -4 `
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`x = - 8`

Therefore,  first term is ` 5x + 2 = 5(-8) + 2 = -40 + 2 = -38`  .
The second term is `7x - 4 = 7(-8) - 4 = -56 - 4 = -60` .
The last term is `10x + 6 = 10(-8) + 6 = -80 + 6 = -74` .
That is it! :)

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