MHB Arithmetic Sequence Confusion // a{n-1} and a{n+1}

[Machara]

So I have this problem I'm stuck on wrapping my head around a particular problem "In the sequence a{n}, let a{0}=2. If a{n+1} = 3 a{n} −1, then what is the value of a3?"

I understand it's following the pattern of each term, and that with Arithmetic sequence a{n-1} means you would use the a{n} term immediately prior (e.g solving for a{2} you would use the result of a{1}) but in this particular problem the sequence is a{n+1} which one would assume you would use the result of the term after? since it's Plus 1 not Minus 1, but that doesn't make sense.

On top of I can elaborate the solutions answer on my worksheet to explain what's happening, and it's saying for a{2} you would input the solution from a{1} to solve for a{2} but wouldn't that be implying the sequence is a{n-1} not a{n+1}?

What's the difference? am I missing something?
This is the elaborated solution that the webpage answers for me:

"Explanation:

To find a{3}
, first find a{1} and a{2}. The sequence says a(n+1)=3a{n}−1, and we know a{0}=2. So, we can find the rest of the sequence, starting with a{1}

.
a{0}=2
a{1}=3(2)−1=5
a{2}=3(5)−1=14
a{3}=3(14)−1=41

So, a{3}=41"

 

[HallsofIvy]

So I have this problem I'm stuck on wrapping my head around a particular problem "In the sequence a{n}, let a{0}=2. If a{n+1} = 3 a{n} −1, then what is the value of a3?"

Do you understand what "a{0}= 2, a{n+1}= 3a{n}- 1" means?

You are told that a{0}= 2. a{1}= a{0+ 1} so a{1}= 3a{0}- 1= 3(2) -1= 5. Then a{2}= a{1+ 1}= 3a{1}- 1= 3(5)- 1= 15- 1= 14. Finally, a{3}= a(2+ 1}= 3a(2)- 1= 3(14)- 1= 42- 1= 4.

I understand it's following the pattern of each term, and that with Arithmetic sequence a{n-1} means you would use the a{n} term immediately prior (e.g solving for a{2} you would use the result of a{1}) but in this particular problem the sequence is a{n+1} which one would assume you would use the result of the term after? since it's Plus 1 not Minus 1, but that doesn't make sense.

On top of I can elaborate the solutions answer on my worksheet to explain what's happening, and it's saying for a{2} you would input the solution from a{1} to solve for a{2} but wouldn't that be implying the sequence is a{n-1} not a{n+1}?

I have no idea where you got "a-1". The problem clearly says a{n+1}

What's the difference? am I missing something?
This is the elaborated solution that the webpage answers for me:

"Explanation:

To find a{3}
, first find a{1} and a{2}. The sequence says a(n+1)=3a{n}−1, and we know a{0}=2. So, we can find the rest of the sequence, starting with a{1}

.
a{0}=2
a{1}=3(2)−1=5
a{2}=3(5)−1=14
a{3}=3(14)−1=41

So, a{3}=41"

I don't see what you could be misunderstanding about this. They took each "a" value, in turn, and calculated 3a(n)- 1 to find the next "a".