Work out the value of x for this arithmetic sequence

[bushra1175]

Hi guys, a little help with this question? It says that x is less than 0 which could be any number <0 ? how do I find out what x is?

 

[yoscar04]

Hi, do you know the definition of an arithmetic sequence?
Start from there and let's see how it goes.

 

[bushra1175]

Hi, do you know the definition of an arithmetic sequence?
Start from there and let's see how it goes.

Yes, an arithmetic sequence increases or decreases by a constant amount. I thought x was 2 and decreasing by 0.5 but then I realised x is supposed to be less than 0. I'm so stuck

 

[yoscar04]

Yes, an arithmetic sequence increases or decreases by a constant amount. I thought x was 2 and decreasing by 0.5 but then I realised x is supposed to be less than 0. I'm so stuck

Write 3 equations for the first three terms of the arithmetic sequence using the definition and then include the data given to you:
a₁=x
a₂=?=1/x
a₃=?=1.
Solve for x, and d, and don't forget that x<0.

 

[pka]

Hi guys, a little help with this question? It says that x is less than 0 which could be any number <0 ? how do I find out what x is?

View attachment 20539
Here is the setup: \(a_1,~a_2=a_1+(1)d,~\&~a_3=a_1+(2)d\)
\(a_1=x\\a_2=x+d=\frac{1}{x}\\a_3=x+2d=1\).

 

[R.M.]

Yes, an arithmetic sequence increases or decreases by a constant amount. I thought x was 2 and decreasing by 0.5 but then I realised x is supposed to be less than 0. I'm so stuck

Using that definition, a₂ - a₁ = a₃ - a₂. Now substitute and solve.

 

[pka]

I substituted a3=x+2d=1 and I got the below, but I'm struggling to solve as I don't know either x or d. I'm really sorry guys, but please tell me how you're working this out?

,

,

We will not work this for you. Please post your work.
I get \(d=\frac{1}{x}-x\) from \(a_2=x+(1)d=\frac{1}{x}\)
Now \(a_3=x+2d=1\) solve for \(x\) (Remember that \(x<0\)).

 

[bushra1175]

We will not work this for you. Please post your work.
I get \(d=\frac{1}{x}-x\) from \(a_2=x+(1)d=\frac{1}{x}\)
Now \(a_3=x+2d=1\) solve for \(x\) (Remember that \(x<0\)).

I got this far, and if I rearrange to get rid of the 1/x I will end up with -x = 1 which doesn't seem right?

 

[yoscar04]

I got this far, and if I rearrange to get rid of the 1/x I will end up with -x = 1 which doesn't seem right?View attachment 20546

Keep going and now solve 1/x-x=1-1/x for x, at the end don't forget that x<0.

 

[bushra1175]

Keep going and now solve 1/x-x=1-1/x for x, at the end don't forget that x<0.

yes I tried to rearrange and I ended up with -x = 1 ( I guess I need to seriously revise rearranging equations). I checked on wolfram alfa and it says x =1 or -2 so I guess the answer is -2. Can you please let me know how to derive this from 1/x-x=1-1/x?

 

[yoscar04]

1/x-x=(1-x²)/x and
1-1/x=(x-1)/x. Therefore
(1-x²)/x=(x-1)/x
x²+x-2=0. Now I expect you to solve this quadratic equation for x.
What do you get? How many answers do you get?

 

[bushra1175]

1/x-x=(1-x²)/x and
1-1/x=(x-1)/x. Therefore
(1-x²)/x=(x-1)/x
x²+x-2=0. Now I expect you to solve this quadratic equation for x.
What do you get? How many answers do you get?

ahh I see. Thank you very much

 

[Steven G]

If two quantities are equal (you know they are equal if there is an equal sign between them), then if you change their signs the new quantities will be equal.

Examples: 3=3 and -3 = -3, -7 =-7 and 7=7.

So if -x=1, then x=-1!

 

[HallsofIvy]

Rather than "change their signs" I would have said "multiply both sides by -1".