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 +...
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.
5 Answers
embizze | High School Teacher | (Level 2) Educator Emeritus
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! :)
If a, b and c are 3 consecutive terms of an arithmetic series, we have a+c = 2*b.
In the problem 5x + 2, 7x - 4 and 10x + 6 are consecutive terms of an arithmetic progression.
This gives: 5x + 2 + 10x + 6 = 2*(7x - 4)
15x + 8 = 14x - 8
x = -16
Substituting x = -16 gives the terms as 5x + 2 = -16*5 + 2 = -80 + 2 = -78, 7x - 4 = 7*-16 - 4 = -112 - 4 = -116 and 10x + 6 = -16*10 + 6 = -160 + 6 = -154
The value of x = -16 and the three terms are -78, -116 and -154
ajasmine14 | Student, Grade 11 | eNotes Newbie
oh i see where you made the mistake was made, in the 2nd equation
2x -d = 6 not 2
ahhh thank you! thank you! you saved me.
ajasmine14 | Student, Grade 11 | eNotes Newbie
i see where you are going. i should have posted the answer to the question. i understand your steps but the answer is incorrect. the answer in the back of my math book is x=-16; the first three terms: -78, -116, -154.