Sequences and Series - Three Consecutive terms given, how to solve for them?

[markl77]

Hi!
The question that was given Is
The terms 5x+2, 7x-4 and 10x+6 are consecutive terms of an arithmetic sequence. Determine the value of x and state the three terms.

My initial work was to solve for x in each of the terms, so:
5x+2
x=-2/5

7x-4
x=4/7

10x+6
x=-6/10

But I found that these weren't consecutive terms; and I don't think that x would have 3 different values.
My second approach was to use the t_n=t₁+d(n-1) but I didn't figure out how to put the numbers in that equation... The only thing that I could think of is converting each of the expressions back into that form, but I have no idea how to do that.

Help is appreciated!!

 

[mmm4444bot]

The terms 5x+2, 7x-4 and 10x+6 are consecutive terms of an arithmetic sequence. Determine the value of x and state the three terms.

My initial work was to solve for x in each of the terms, so:
5x+2
x=-2/5

7x-4
x=4/7

10x+6
x=-6/10

You cannot solve for a variable, unless you have an equation.

In other words, you first "decided" that each of the three terms was zero, and then you solved each of the following equations.

5x + 2 = 0
x = -2/5

7x - 4 = 0
x = 4/7

10x + 6 = 0
x = -6/10

You're correct in realizing that x cannot represent three different values, here, so the trivial arithmetic sequence ...0,0,0,... cannot be correct.

You know that each term in an arithmetic sequence is found by adding the common difference to the preceding term, yes?

In other words, if you subtract a preceding term from the term which follows it, then you get the common difference.

How about you try that? You're given three consecutive terms. If you subtract the first term from the second, you will get the common difference.

Adding that common difference to the second term gives you the third term -- and that yields an equation that you can solve for x.

If you need more help, please show what you tried or explain what you're not sure about.

 

[markl77]

You cannot solve for a variable, unless you have an equation.

In other words, you first "decided" that each of the three terms was zero, and then you solved each of the following equations.

5x + 2 = 0
x = -2/5

7x - 4 = 0
x = 4/7

10x + 6 = 0
x = -6/10

You're correct in realizing that x cannot represent three different values, here, so the trivial arithmetic sequence ...0,0,0,... cannot be correct.

You know that each term in an arithmetic sequence is found by adding the common difference to the preceding term, yes?

In other words, if you subtract a preceding term from the term which follows it, then you get the common difference.

How about you try that? You're given three consecutive terms. If you subtract the first term from the second, you will get the common difference.

Adding that common difference to the second term gives you the third term -- and that yields an equation that you can solve for x.

If you need more help, please show what you tried or explain what you're not sure about.

Thankyou.

I tried what you said, since we know that 7x-4 comes after 5x+2, then I can just subtract 5x+2 from 7x-4

So my work for that Is:

7x-4-(5x+2)
=7x-4-5x-2
=2x-6. So 2x-6 is the Common Difference?

But then since 7x-4 and 10x+6 are also consecutive, shouldn't I get the same Common Difference?

10x+6-(7x-4)
=10x+6-7x+4
=3x+10

I don't know which Common Difference to use, maybe I did the math incorrectly?

 

[mmm4444bot]

I tried what you said, since we know that 7x-4 comes after 5x+2, then I can just subtract 5x+2 from 7x-4

So my work for that Is:

7x-4-(5x+2)
=7x-4-5x-2
=2x-6. So 2x-6 is the Common Difference?

2x - 6 is an expression that represents the Common Difference.
The actual Common Difference is a number, not an expression.

But then since 7x-4 and 10x+6 are also consecutive, shouldn't I get the same Common Difference?

You will get a different expression that also represents the Common Difference.

10x+6-(7x-4)
=10x+6-7x+4
=3x+10

Your work is correct, so far.

Now, I was thinking that you could form an equation to solve by adding 2x-6 to 7x-4 and setting that equal to 10x+6.

But you have stumbled upon a different equation to solve, instead!

In other words, you have found two different expressions which each represent the Common Difference; hence, these expressions must equal one another.

2x - 6 = 3x + 10

Solve this equation, to find x. Then you'll have everything that you need, to determine the actual value of the Common Difference, as well as the value of each of the three consecutive terms. :cool:

 

[HallsofIvy]

Thankyou.

I tried what you said, since we know that 7x-4 comes after 5x+2, then I can just subtract 5x+2 from 7x-4

So my work for that Is:

7x-4-(5x+2)
=7x-4-5x-2
=2x-6. So 2x-6 is the Common Difference?

But then since 7x-4 and 10x+6 are also consecutive, shouldn't I get the same Common Difference?

Yes, you should! That is the meaning of common difference.

10x+6-(7x-4)
=10x+6-7x+4
=3x+10

I don't know which Common Difference to use, maybe I did the math incorrectly?

It is isn't a matter of "which" to use- to be a Common Difference they must be the same:
2x- 6= 3x+ 10. NOW, what is x?