Why is Sequence 1, 4, 7, 10 Written as 3n-2 or 3n+2?

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In summary: So just be aware of that when you're working with a formula like a_n = 3n-2, which assumes counting from 1.In summary, the conversation discusses the use of different forms for a sequence, specifically 3n-2 and 3n+2, and the difference in starting points for counting (0 or 1). It is noted that starting with 0 can sometimes lead to confusion and that it ultimately depends on personal preference. Additionally, it is mentioned that some textbooks may use a different starting point for counting.
  • #1
Natasha1
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Hi,

Could anyone please explain to me very simply why in a sequence say 1, 4, 7, 10,... which has the general term form: a_n = 3n-2 can also be written as 3n+2?

Why do some people use 3n+2 rather than 3n-2, what advantage has that got? Is it actually more mathematically correct to write 3n+2?

So 2n-2 or 2n+2 would be for a sequence 1, 2, 3, 4, 5,...
4n-1 or 4n+3 would be for 3, 7, 11, ...

Thanks

Nat
 
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  • #2
Surely you mean 3n+1 with n starting from 0??
 
  • #3
If an= 3n-2 then 3(n+1)- 2= 3n+3-2= 3n+1. The only difference is that with an= 3n-2 you have a1= 1, a2= 4, etc. while with an= 3n+1 it is a0= 1, a1= 4, etc. Just a difference in where you start counting.
There is no real advantage- just that some people don't like to start counting with 0!


"So 2n-2 or 2n+2 would be for a sequence 1, 2, 3, 4, 5,... "

No- 2(1)-2= 0, 2(2)-2= 2, but 2(3)- 2= 4 not 3. It should be obvious that 2n- 2 and 2n+ 2 are always even numbers. Did you mean 2, 4, 6, ...?


"4n-1 or 4n+3 would be for 3, 7, 11, ..."
Yes, one starts with n= 1, the other with n= 0.
 
  • #4
HallsofIvy said:
If an= 3n-2 then 3(n+1)- 2= 3n+3-2= 3n+1. The only difference is that with an= 3n-2 you have a1= 1, a2= 4, etc. while with an= 3n+1 it is a0= 1, a1= 4, etc. Just a difference in where you start counting.
There is no real advantage- just that some people don't like to start counting with 0!


"So 2n-2 or 2n+2 would be for a sequence 1, 2, 3, 4, 5,... "

No- 2(1)-2= 0, 2(2)-2= 2, but 2(3)- 2= 4 not 3. It should be obvious that 2n- 2 and 2n+ 2 are always even numbers. Did you mean 2, 4, 6, ...?


"4n-1 or 4n+3 would be for 3, 7, 11, ..."
Yes, one starts with n= 1, the other with n= 0.

Yes sorry I did mean 2, 4, 6, 8, ...
 
  • #5
I don't know about HallsofIvy, but I'm an inveterate "from zero"-counter..
 
Last edited:
  • #6
arildno said:
I don't know about HallsofIvy, but I'm an inveterate "from zero"-counter..

Unfortunately (or fortunately depending on your preference), many textbooks count from 1.
 

1. Why is the sequence 1, 4, 7, 10 written as 3n-2 or 3n+2?

The sequence 1, 4, 7, 10 is written as 3n-2 or 3n+2 because it follows a pattern where each number is 3 more than the previous number. This can be represented as 3n, where n is the position of the number in the sequence (starting at 0). However, since the first number in the sequence is 1 and not 0, we need to subtract 2 from 3n to get the correct sequence. Thus, the formula becomes 3n-2.

2. What does the "n" stand for in the formula 3n-2 or 3n+2?

The "n" in the formula 3n-2 or 3n+2 represents the position of the number in the sequence. For example, in the sequence 1, 4, 7, 10, the first number (1) is in position 0, the second number (4) is in position 1, and so on. The "n" allows us to calculate any number in the sequence by plugging in its position.

3. How do you know if the formula for a sequence is 3n-2 or 3n+2?

The formula for a sequence is 3n-2 or 3n+2 if the pattern of the sequence follows the rule of each number being 3 more than the previous number. To determine whether it is 3n-2 or 3n+2, you can look at the first few numbers in the sequence and see if they match the formula. If the first number is 1, it is likely 3n-2, and if the first number is 2, it is likely 3n+2.

4. Can this formula be used for any sequence?

No, this formula can only be used for sequences that follow the pattern of each number being 3 more than the previous number. Other sequences may have different patterns and therefore, require different formulas to calculate the numbers in the sequence.

5. What is the purpose of writing a sequence in the form of 3n-2 or 3n+2?

Writing a sequence in the form of 3n-2 or 3n+2 allows us to easily calculate any number in the sequence by plugging in its position. It also helps us to identify and understand the pattern of the sequence, making it easier to predict the next numbers in the sequence. This can be useful in various fields of science, such as mathematics, physics, and computer science.

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