In the study of sequences, understanding how to derive explicit formulas is essential. An explicit formula allows you to calculate the terms of a sequence based on their position, denoted by \( n \). The process often involves identifying patterns within the sequence of numbers.

One common pattern is an arithmetic sequence, where the terms increase by a constant amount. For example, in the sequence 5, 6, 7, 8, 9, each term increases by 1. The general formula for such sequences can be expressed as \( a_n = n + c \), where \( c \) is a constant that adjusts the starting point. In this case, since the first term is 5, we find that \( c = 4 \), leading to the formula \( a_n = n + 4 \). This can be verified by substituting values of \( n \) to ensure the correct terms are produced.

Another type of sequence involves alternating signs, such as -5, 5, -5, 5. This pattern can be represented using \( (-1)^n \), which alternates between positive and negative values based on whether \( n \) is even or odd. To adjust for the specific values in the sequence, we multiply by 5, resulting in the formula \( a_n = (-1)^n \cdot 5 \). This ensures that the first term is -5 and the second term is 5, continuing the pattern.

Sequences can also include fractions, such as \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} \). Here, the numerators follow the pattern of \( n \), while the denominators increase by 1 starting from 2. Thus, the general formula for this sequence is \( a_n = \frac{n}{n + 1} \), which can be confirmed by substituting values of \( n \) to match the sequence.

Lastly, exponential sequences, like 2, 4, 8, 16, 32, grow at an increasing rate. The general formula for such sequences is expressed as \( a_n = b^n \), where \( b \) is the base of the exponential growth. In this case, since the sequence consists of powers of 2, the formula is \( a_n = 2^n \). This can be validated by substituting values of \( n \) to ensure the correct terms are generated.

By recognizing these patterns—arithmetic, alternating signs, fractional, and exponential—you can effectively derive explicit formulas for various sequences. This skill is crucial for solving problems related to sequences and series in mathematics.