Problem 24 - Calculus for Biology and Medicine

Question

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ 1,3,9,27,81 $$

Step-by-Step Solution

Answer

The next four values are 243, 729, 2187, and 6561.

Step 1: Identify the Pattern

Examine the given sequence: 1, 3, 9, 27, 81. Notice how each term is obtained by multiplying the previous term by 3. This means it's a geometric sequence where the common ratio \(r\) is 3.

Step 2: Verify Pattern

Starting with \(a_0 = 1\), multiply by 3 to get \(a_1 = 3\), then \(a_1 \times 3 = a_2 = 9\), \(a_2 \times 3 = a_3 = 27\), and \(a_3 \times 3 = a_4 = 81\). This confirms the recurrence relationship \(a_n = 3\times a_{n-1}\).

Step 3: Calculate Next Term (a_5)

Using the pattern \(a_4 \times 3 = a_5\), calculate \(81 \times 3 = 243\). So, \(a_5 = 243\).

Step 4: Calculate Second Next Term (a_6)

Apply the pattern to determine \(a_6\): \(a_5 \times 3 = 243 \times 3 = 729\). So, \(a_6 = 729\).

Step 5: Calculate Third Next Term (a_7)

Following the same steps, find \(a_7\): \(a_6 \times 3 = 729 \times 3 = 2187\). Hence, \(a_7 = 2187\).

Step 6: Calculate Fourth Next Term (a_8)

Finally, calculate \(a_8\) using the pattern: \(a_7 \times 3 = 2187 \times 3 = 6561\). Therefore, \(a_8 = 6561\).

Key Concepts

Common Ratio

In a geometric sequence, each term after the first is obtained by multiplying the previous term by a constant called the **common ratio**. This ratio remains the same throughout the entire sequence. The sequence given in the exercise, 1, 3, 9, 27, 81, clearly demonstrates this with a common ratio of 3.

Simply put, when each term is divided by its preceding term, the result is always the same. For instance:

• 3 divided by 1 equals 3
• 9 divided by 3 equals 3
• 27 divided by 9 equals 3
• 81 divided by 27 equals 3

This repeating factor is what maintains the uniformity of the sequence and facilitates the calculation of subsequent terms.
Understanding the **common ratio** is crucial because it helps identify the type of sequence and guides the development of the **recurrence relationship**.

Recurrence Relationship

The **recurrence relationship** is a formula that expresses each term in the sequence based on the preceding term using the **common ratio**. In the sequence 1, 3, 9, 27, 81, this relationship is symbolized as:\[ a_n = 3 \times a_{n-1} \] This formula indicates that each term, \( a_n \), is derived by multiplying the previous term, \( a_{n-1} \), by 3.

The usefulness of a **recurrence relationship** lies in its simplicity—it allows us to compute all future terms of the sequence by just knowing the present term and the **common ratio**. It serves as the backbone for determining the sequence pattern and further simplifies term calculation.

Recognizing the recurrence relationship makes the calculation of future terms systematic and conflict-free.

Sequence Pattern

The **sequence pattern** in a geometric sequence is driven by the consistent application of the recurrence relationship based on the common ratio. In simpler terms, it's about recognizing that the sequence progresses by multiplying each term steadily by the same number. In our example:

• From 1 to 3, 3 is multiplied
• 3 to 9 continues with multiplying by 3
• 9 to 27, you multiply by 3 again
• This pattern continues for each successive term

Understanding the sequence pattern means recognizing the multiplicative factor at every step, which ensures the sequence is geometric, predicting future terms effortlessly. The uniformity of this pattern is the hallmark of geometric sequences and aids in confirming the recurrence relationship.

Term Calculation

**Term calculation** in a geometric sequence is all about applying the common ratio to find future terms effectively. Armed with the recurrence relationship and understanding of the sequence pattern, each successive term can be computed quickly.

Reusable formula:
\[ a_n = r \times a_{n-1} \]where \( r \) is the common ratio.

Let's illustrate that:

• Start from \( a_0 = 1 \)
• Calculate \( a_1 = 3 \times 1 = 3 \)
• Determine \( a_2 = 3 \times 3 = 9 \)
• Extend to \( a_3 = 3 \times 9 = 27 \)
• Continue to \( a_4 = 3 \times 27 = 81 \)

This same method allows us to calculate further terms:

• \( a_{5} = 243 \)
• \( a_{6} = 729 \)
• \( a_{7} = 2187 \)
• \( a_{8} = 6561 \)

Mastering term calculation through the recurrence relationship primes users to handle larger sequences with efficiency and builds confidence in the mechanics of sequence progression.