Question

Write a recursive rule for the sequence. $$ 44,11, \frac{11}{4}, \frac{11}{16}, \frac{11}{64}, \ldots $$

Short answer

The recursive rule for the sequence is: \(F(1) = 44\) and \(F(n) = 0.25 * F(n-1)\) for \(n > 1\).

Step-by-step solution

- Identify the Common Ratio

Evaluate the ratio of any term to the previous term in the sequence. Here, the second term (11) divided by the first term (44) is 0.25. The third term \(\frac{11}{4}\) divided by the second term (11) is also 0.25. As there is a consistent result, the common ratio, r, is 0.25. In other words, we are multiplying each term by 0.25 to get the next term.

- Formulate the Recursive Rule

A recursive rule for a sequence tells you the term which preceded the current one. The rule thus for our sequence is the following: \(F(1) = 44\) and \(F(n) = 0.25 * F(n-1)\) for \(n > 1\). This means the first term is 44 and each following term is 0.25 times the previous term.

Key concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which you multiply each term to get the next one. It's a fundamental part of understanding how the sequence progresses.
For the sequence 44, 11, \(\frac{11}{4}\), \(\frac{11}{16}\), ... you calculate the common ratio by dividing any term by the one before it.

• From 44 to 11, divide: \(\frac{11}{44} = 0.25\).
• From 11 to \(\frac{11}{4}\), divide: \(\frac{11/4}{11} = 0.25\).

This constant 0.25 tells us we repeatedly multiply by 0.25 to get from one term to the next. Recognizing this factor is crucial in both understanding and constructing geometric sequences.

Geometric Sequences

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This consistent multiplication makes the sequence predictable.
An example sequence is: 44, 11, \(\frac{11}{4}\), \(\frac{11}{16}\), ...
How does it work?

• Start with a first term: here, 44.
• Multiply by the common ratio (0.25) to get the next: \(44 \times 0.25 = 11\).
• Repeat the process: \(11 \times 0.25 = \frac{11}{4}\).

Geometric sequences are widely used in different fields, showing exponential growth or decay patterns. Recognizing them helps in solving complex problems more easily.

Recursive Rule

A recursive rule in mathematics provides a way to express each term of a sequence using the preceding terms. It's like following a recipe step by step, where each step depends on the one before.
For our sequence, the recursive rule is:

• Start with the first term: \(F(1) = 44\).
• Define the rule for the following terms: \(F(n) = 0.25 \times F(n-1)\) for \(n > 1\).

This means after the first term, each term is 0.25 times the previous term.
Understanding recursive rules allows you to extend sequences indefinitely without calculating each step from scratch. They're especially useful in programming and mathematical proofs where processes need to be repeated.