Arithmetic Sequence Homework: Find x for Consecutive Terms

[Faiien]

Homework Statement

Find x so that x+5, 3x+1, and 4x+1 are consecutive terms of an arithmetic sequence.

Not really sure how to do the problem at all. Some assistance would be much appreciated.

 

[Hurkyl]

What do you know about "three consecutive terms in an arithmetic sequence"?

 

[Faiien]

Because it's an arithmetic sequence, if you add a constant to one solution you can get the next solution.

 

[Hurkyl]

Because it's an arithmetic sequence, if you add a constant to one solution you can get the next solution.

"solution"? Did you mean "term"?

What do you know about three consecutive terms in a sequence?



P.S. this is algebra. Name things with variables. Express facts as equations.

 

[Faiien]

Yes, I apologize, I did mean terms.
Because the terms of an arithmetic sequence always vary by a constant
x+a, x+2a. x+3a
you can attain a, the constant, by subtracting two consecutive terms.
(x+2a)-(x+a)=a or (x+3a)-(x+2a)=a

 

[Hurkyl]

Yes, I apologize, I did mean terms.
Because the terms of an arithmetic sequence always vary by a constant
x+a, x+2a. x+3a
you can attain a, the constant, by subtracting two consecutive terms.
(x+2a)-(x+a)=a or (x+3a)-(x+2a)=a

Let's not use 'x' here! We're already using 'x' for something else!

So we know "three consecutive terms of an arithmetic sequence" can be written as

r+a, r+2a, r+3a​

for an appropriate choice of r and a.

So now, how do you express the fact that

x+5, 3x+1, and 4x+1 are "three consecutive terms of an arithmetic sequence"​

?