In this example, we explore how to derive a general formula for a sequence based on its first four terms. The sequence provided is:

1/1×2, 1/2×3, 1/3×4, 1/4×5.

To find the general term, denoted as \( a_n \), we first observe the pattern in the sequence. Notably, each term is a fraction, indicating that our general formula will also involve a fraction. The numerators of all terms are consistently 1, which suggests that the numerator in our formula will simply be 1, as it does not depend on \( n \).

Next, we analyze the denominators, which are products of two numbers: for the first term, it is \( 1 \times 2 \); for the second, \( 2 \times 3 \); for the third, \( 3 \times 4 \); and for the fourth, \( 4 \times 5 \). The first number in each denominator corresponds directly to the term's index \( n \), indicating that the first part of the denominator is \( n \).

For the second number in the denominator, we see a pattern where it increases by 1 with each term: 2, 3, 4, and 5. This suggests that the second number can be represented as \( n + 1 \). Therefore, the complete denominator can be expressed as \( n(n + 1) \).

Combining these observations, we arrive at the general formula for the sequence:

\( a_n = \frac{1}{n(n + 1)} \).

To find the 15th term of the sequence, we substitute \( n = 15 \) into our formula:

\( a_{15} = \frac{1}{15(15 + 1)} = \frac{1}{15 \times 16} = \frac{1}{240} \).

Calculating this gives approximately \( 0.0041667 \), which can be rounded to \( 0.0042 \). Thus, the 15th term in the sequence is \( 0.0042 \).

This process illustrates how to identify patterns in sequences and derive a general formula, allowing for quick calculations of specific terms without needing to list all preceding terms.