Problem 19 - Algebra 2

Question

Write an explicit formula for each sequence. Then find \(a_{12}\) $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \dots $$

Step-by-Step Solution

Answer

The explicit formula is \(a_n = 1/(n+1)\) and the 12th term of the sequence, \(a_{12}\), is \(1/13\).

Step 1: Identify the pattern

The given sequence is decreasing, and it seems like the nth term can be represented by the formula \(1/(n+1)\). We can identify that for each term the denominator is one number higher than the position of the term.

Step 2: Write out the explicit formula

Given the pattern spotted, the explicit formula for the sequence will be \(a_n = 1/(n+1)\).

Step 3: Find the 12th term

Substitute \(n = 12\) in the explicit formula. So \(a_{12} = 1/(12 + 1) = 1/13\)

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression used to find any term in a sequence without having to refer to previous terms. Instead of calculating each term one by one, you immediately arrive at the answer by substituting the term's position directly in the formula.

In the exercise above, the explicit formula helps to simplify and speed up the process of finding any term in the sequence. The sequence given is based on a pattern where each term decreases as the denominator increases. We can observe that the position of any term, denoted as \(n\), directly influences its denominator.

• For the first term when \(n = 1\), the term is \(\frac{1}{2}\).
• For the second term (n = 2), it's \(\frac{1}{3}\).
• This leads us to a general explicit formula of \(a_n = \frac{1}{n+1}\).

By using this formula, we can confidently find any term in the sequence by merely knowing its position \(n\). This method is particularly advantageous when dealing with large sequences or when only specific terms are of interest.

Term Calculation

Once we have an explicit formula, calculating specific terms in a sequence becomes a straightforward task. This process is known as term calculation. Using the formula \(a_n = \frac{1}{n+1}\), the term can be calculated by substituting \(n\) with the desired term number.

In our example, we are tasked with finding the 12th term of the sequence. This involves performing a simple substitution:

• Plug \(n = 12\) into the explicit formula \(a_n = \frac{1}{n+1}\).
• The calculation becomes \(a_{12} = \frac{1}{12+1}\).
• Therefore, \(a_{12} = \frac{1}{13}\).

Term calculation using an explicit formula is efficient because it eliminates repetitive addition or multiplication steps present in recursive sequences. By understanding and applying the explicit formula, you can quickly determine any term within the given sequence.

Pattern Recognition

Recognizing patterns in sequences is a fundamental step towards developing an explicit formula. Pattern recognition involves observing how sequences progress to identify a consistent rule or relationship between terms. In practice, this often involves noting changes amongst consecutive terms.

• In the given sequence \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \), each term's denominator is one more than the term's position (starting from \(n=1\)).
• For instance, the pattern "denominator equals the position number plus one" can be inferred straight from observing the first few terms.

Identifying this pattern helps construct the explicit formula \(a_n = \frac{1}{n+1}\), which describes the entire sequence accurately. Recognizing such patterns in algebra sequences equips you with the ability to derive formulas, simplifying complex problems. This skill is crucial as it goes beyond sequential calculation and enhances analytical abilities in mathematics.