Problem 94 - Calculus for Biology and Medicine

Question

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=-2 a_{n}, a_{0}=2 $$

Step-by-Step Solution

Answer

The sequence terms are: \(a_{1} = -4, a_{2} = 8, a_{3} = -16, a_{4} = 32, a_{5} = -64\).

Step 1: Understand the Recurrence Relation

The sequence is defined with a recurrence relation: \(a_{n+1} = -2a_{n}\), with the initial term \(a_{0} = 2\). This means each term is generated by multiplying the previous term by \(-2\).

Step 2: Calculate \(a_{1}\)

To find \(a_{1}\), use the formula \(a_{n+1} = -2a_{n}\) starting with \(n = 0\). This gives \(a_{1} = -2 \times a_{0} = -2 \times 2 = -4\).

Step 3: Calculate \(a_{2}\)

Now, to find \(a_{2}\), apply the formula again: \(a_{2} = -2 \times a_{1} = -2 \times (-4) = 8\).

Step 4: Calculate \(a_{3}\)

Use the recurrence relation once more to find \(a_{3}\): \(a_{3} = -2 \times a_{2} = -2 \times 8 = -16\).

Step 5: Calculate \(a_{4}\)

Continue with the formula to find \(a_{4}\): \(a_{4} = -2 \times a_{3} = -2 \times (-16) = 32\).

Step 6: Calculate \(a_{5}\)

Finally, determine \(a_{5}\): \(a_{5} = -2 \times a_{4} = -2 \times 32 = -64\).

Key Concepts

Understanding the Recursive Sequence

When working with a recursive sequence, we encounter a series of numbers where each term is determined based on one or more of its preceding terms. This type of sequence is defined using a recurrence relation. In our specific exercise, the sequence is formed with the relation \(a_{n+1} = -2a_{n}\). This relation tells us how to compute each subsequent term from the previous one.

Recurrence relations are the backbone of recursive sequences. They provide a formulaic approach to generate terms sequentially. In this particular example, the relation suggests that to find the next term, you multiply the previous term by \(-2\). This results in alternately positive and negative numbers, a feature that can often be seen in recursive sequences.

In real-world scenarios or more advanced mathematical contexts, recursive sequences can become complex and involve more variables, but the core idea of relying on previous terms to define future ones remains constant.

Role of Initial Condition

The initial condition in recursive sequences acts as the starting point from which all calculations begin. It is crucial because, without it, our sequence has no starting value and cannot progress. In the original exercise, the initial condition is given as \(a_{0} = 2\). This means our sequence starts with the term \(2\).

Initial conditions are foundational. They ground the sequence and ensure that every subsequent term is reliable and reflects the intended pattern or series behavior. By specifying \(a_{0} = 2\), the sequence develops in a very predictable and orderly manner. The value of 2 acts as a "seed" from which all other terms can be accurately computed using the recurrence relation.

Step-by-Step Term Calculation

The computation of each term in a recursive sequence is dependent on the term immediately preceding it. Let's break down the process for the first few terms based on our exercise's recurrence relation \(a_{n+1} = -2a_{n}\) and initial condition:

• **Calculate \(a_{1}\):** Start with \(n = 0\). Using the relation, we find \(a_{1} = -2 \times 2 = -4\).
• **Calculate \(a_{2}\):** Proceed with the next term using \(a_{1}\): \(a_{2} = -2 \times (-4) = 8\).
• **Calculate \(a_{3}\):** Continue with \(a_{2}\): \(a_{3} = -2 \times 8 = -16\).
• **Calculate \(a_{4}\):** Moving forward with \(a_{3}\): \(a_{4} = -2 \times (-16) = 32\).
• **Calculate \(a_{5}\):** Finally, from \(a_{4}\): \(a_{5} = -2 \times 32 = -64\).

Each step builds upon the former, ensuring the sequence remains connected and reflects its recursive nature. By consistently applying the given relation, each term is effectively and accurately calculated.