Problem 14 - Algebra 2

Question

Write a recursive formula for each sequence. Then find the next term. $$ 40,20,10,5, \frac{5}{2}, \dots $$

Step-by-Step Solution

Answer

The recursive formula for the sequence is \(a_1 = 40\), \(a_n = a_{n-1}/2\) for \(n > 1\). The next term in the sequence is \(\frac{5}{4}\).

Step 1: Identify The Pattern

One important part of defining a recursive sequence is identifying the pattern of the sequence. In this sequence \(40, 20, 10, 5, \frac{5}{2}\), each term is half of the previous one, because \(20 = \frac{40}{2}\), \(10 = \frac{20}{2}\), \(5 = \frac{10}{2}\), and \(\frac{5}{2} = \frac{5}{2}\).

Step 2: Write A Recursive Formula

Next, we need to describe this pattern mathematically with a recursive formula. The first term of the sequence is \(40\), and each following term is half of the previous term. The recursive formula is thus \(a_1 = 40\), \(a_n = a_{n-1}/2\) for \(n > 1\). Where \(a_n\) represents the nth term and \(a_{n-1}\) is the term before it.

Step 3: Find The Next Term

To find the next term, rewrite the formula inserting the position of the next term. The next term after \(\frac{5}{2}\) is the 6th term (\(a_6\)). So, \(a_6 = a_{5}/2 = \frac{5}{2} / 2 = \frac{5}{4}\). Therefore, the sixth term of the sequence is \(\frac{5}{4}\).

Key Concepts

Geometric Sequences

Geometric sequences are mathematical patterns where each term is determined by multiplying (or dividing, which is essentially multiplying by a fraction) the previous term by a constant value. This constant is known as the "common ratio." In the given sequence, 40, 20, 10, 5, \( \frac{5}{2} \), you can see that each term is derived by dividing the previous term by 2. This is equivalent to multiplying the term by a common ratio of \( \frac{1}{2} \).

• Starting value: 40
• Common Ratio: \( \frac{1}{2} \)

Geometric sequences appear in various mathematical and real-world contexts, such as calculating compound interest, analyzing population growth, and measuring radioactive decay. Recognizing these sequences can help simplify complexities into a recognizable pattern of growth or decay.

Recursive Formulas

A recursive formula allows us to express each term of a sequence based on the preceding term(s). It creates a chain-like progression, where knowing one piece helps to find the next. The beauty of recursive formulas is in their simplicity—by defining a starting point and a repetitive rule, the sequence unfolds step by step.

In the given problem, the recursive formula identifies the first term as 40 and describes subsequent terms as half of their predecessor. Mathematically, this formula is denoted as:

• First term: \(a_1 = 40\)
• Recursive rule: \(a_n = \frac{a_{n-1}}{2}\) for \(n > 1\)

Using this recursive rule, you can calculate any term in the sequence provided you have the preceding term. This method is practical for sequences with a clear pattern or rule.

Sequence Patterns

Understanding sequence patterns is crucial for generating and predicting future terms in any sequence. A sequence pattern is the rule or relationship that defines how to transition from one term to the next.

• Identifying a sequence pattern involves observing how terms change from one to another.
• In mathematical problems, finding this pattern often involves operations like multiplication, division, addition, or subtraction.

In our example, the pattern is simple division by 2 for each next term. Observing that 20 is \( \frac{40}{2} \), 10 is \( \frac{20}{2} \), and so on, highlights this rule.
By understanding sequence patterns, we can write formulas and solve for unknowns, making it a powerful tool in mathematics and beyond.