Question

Find the explicit formula of a sequence.

Answer 1

To find an explicit formula of a sequence, one must understand the pattern of the series, apply proper mathematical theorems such as the binomial theorem if applicable, and list all known values to select the best equations to solve the problem.

Step-by-step explanation:

The student has asked to find an explicit formula of a sequence. In mathematics, series expansions and finding explicit formulas form a critical part of understanding sequences and series.

The binomial theorem is one method that illuminates the expansion of expressions raised to a power and is often used in conjunction with series to solve complex problems. Referring to the series expansions, we can analyze patterns using variables a and b, indices, and factorials to handle the terms of the expansion.

For instance, to find an explicit formula we would evaluate the pattern of the sequence, determine the type of series it forms (arithmetic, geometric, etc.), and use known formulas or create one that fits the sequence by incorporating all known values. Utilizing the binomial theorem or other relevant methods may be necessary depending on the complexity of the sequence.

To apply this effectively, knowing all values and what needs to be solved for is imperative. A step-by-step approach, complemented by tables or other visual aids, helps in setting out known values and in the identification of relevant equations, which are best to use toward finding the explicit formula of a given sequence.

Answer 2

The explicit formula for a sequence represents a mathematical expression that directly gives the nth term of the sequence without referring to earlier terms. It's typically denoted as
`\(a_n = f(n)\),` where
`\(a_n\)` signifies the nth term and
`\(f(n)\)` denotes the formula that generates the terms of the sequence.

Step-by-step explanation:

In mathematics, finding the explicit formula of a sequence involves deriving a formula that directly calculates any term of the sequence based on its position without needing the previous terms. For instance, consider an arithmetic sequence where each term increases by a constant difference from the preceding term. The explicit formula for an arithmetic sequence is
`\(a_n = a_1 + (n-1) \`times d\), where
`\(a_n\)`represents the nth term,
`\(a_1\)`is the first term, and
`\(n\)` denotes the term's position in the sequence.

Suppose we have an arithmetic sequence starting with
`\(a_1 = 3\)` and a common difference of
`\(d = 4\).` To find the explicit formula for this sequence, we use the formula
`\(a_n = a_1 + (n-1) \`times d\). Plugging in the given values, the explicit formula becomes
`\(a_n = 3 + (n-1) \`times
`4 = 3 + 4n - 4 = 4n - 1\)`. Therefore, the explicit formula for this arithmetic sequence is
`\(a_n = 4n - 1\)`.

The explicit formula simplifies the process of determining any term within a sequence without having to calculate each preceding term. It's a concise representation that facilitates quick calculations for specific positions in the sequence.