An arithmetic sequence is defined as a sequence of numbers in which the difference between consecutive terms is constant. To find the nth term of an arithmetic sequence, we typically use the formula:
a(n) = a(1) + (n – 1) * d
Where:
- a(n) is the nth term of the sequence.
- a(1) is the first term of the sequence.
- d is the common difference between consecutive terms.
- n is the term number.
Let’s apply this to the given sequence: 1, 2, 5, 8.
First, identify the first term:
- a(1) = 1
Next, we need to find the common difference d. We calculate the differences between consecutive terms:
- 2 – 1 = 1
- 5 – 2 = 3
- 8 – 5 = 3
The differences are 1, 3, and 3. Since the difference does not remain constant (1, 3, 3), this sequence is not an arithmetic sequence in the traditional sense.
However, we can still derive a formula to express the nth term. To do this, we can observe how the terms increase:
- From 1 to 2, it increases by 1
- From 2 to 5, it increases by 3
- From 5 to 8, it increases by 3
Notice a pattern: the first term increases linearly but changes in the rate of increase subsequently. Given this observation, we can try to formulate an nth term equation based on the distinct behavior of the sequence.
Let’s analyze the position of each number to find a pattern:
- For n=1: a(1) = 1
- For n=2: a(2) = 2
- For n=3: a(3) = 5
- For n=4: a(4) = 8
Now we can suggest an alternative approach. By examining the sequence:
- At n=1, it’s 1
- At n=2, it’s 2
- At n=3, we see a significant jump to 5
- Finally, at n=4, it increments to 8
Using this, we can express the nth term based on the position:
a(n) = 1 + (n – 1) for n = 1
a(n) = 2 + 3(n – 2) for n ≥ 2.
So, while the direct arithmetic formula does not fit here, you can derive things mathematically to describe the nth behavior consistently through understanding its progressive changes.
In summary, expressing this series as a piecewise function can also be a legitimate way to describe its nth term accurately:
a(n) =
- 1, if n = 1
- 2, if n = 2
- 3 + 2(n – 3), if n ≥ 3
This output will satisfy the desired format of describing any term for this series given its unique structure.