To find the nth term of the given arithmetic sequence (1, 2, 5, 8), we first need to identify the pattern of the sequence. An arithmetic sequence is one in which the difference between consecutive terms is constant.
Let’s look at the differences between the terms:
- 2 – 1 = 1
- 5 – 2 = 3
- 8 – 5 = 3
From this, we can see that the differences are not constant. Thus, this sequence is not a traditional arithmetic sequence, but it still allows us to derive a general formula.
To create a formula for this sequence, let’s list the terms again:
- 1 (for n=1)
- 2 (for n=2)
- 5 (for n=3)
- 8 (for n=4)
It is clear that the sequence is not equally spaced in simple arithmetic terms. However, we can observe the way the terms grow. The first two terms increase by 1, while the next terms increase by 3.
This insight suggests that we might define the sequence with a polynomial function instead. A simple guess would be to use the formula:
an = 1 + (n – 1)(1 + 3(n – 2)/2)
After simplification, we can express the nth term more clearly:
an = 1 + (n – 1)(1 + 3/2(n – 2))
Thus, finding a pattern in this unconventional progression leads us to derive the nth-term formula. This formula will allow us to calculate any term in the sequence, regardless of how unusual its pattern appears.