Problem 28 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{1}=3 \text { and } a_{n}=n+2 a_{n-1} \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Question: Find the first five terms of the recursively defined sequence given by the initial term \(a_{1}=3\) and the recursive relationship \(a_{n}=n+2 a_{n-1}\) for \(n \geq 2\). Answer: The first five terms of the sequence are \(a_1=3, \quad a_2=8, \quad a_3=19, \quad a_4=42, \quad a_5=89\).

Step 1: Write initial term

The given initial term is \(a_{1}=3\). This will be the first term of our sequence.

Step 2: Find the second term

Using the recursive relationship formula \(a_{n}=n+2 a_{n-1}\), we can calculate the second term (\(a_2\)) as follows: $$a_{2}=2+2a_{1}=2+2(3)=2+6=8$$ So, the second term of our sequence is 8.

Step 3: Find the third term

Similarly, we can find the third term (\(a_3\)) using the recursive relationship formula: $$a_{3}=3+2a_{2}=3+2(8)=3+16=19$$ The third term of our sequence is 19.

Step 4: Find the fourth term

We continue the process to find the fourth term (\(a_4\)): $$a_{4}=4+2a_{3}=4+2(19)=4+38=42$$ The fourth term of our sequence is 42.

Step 5: Find the fifth term

Finally, we can find the fifth term (\(a_5\)) using the same formula: $$a_{5}=5+2a_{4}=5+2(42)=5+84=89$$ The fifth term of our sequence is 89.

Step 6: Write down the first five terms

Putting it all together, the first five terms of our recursively defined sequence are: $$a_1=3, \quad a_2=8, \quad a_3=19, \quad a_4=42, \quad a_5=89$$

Key Concepts

Recursive Formulas

Recursive formulas are a way to define sequences based on previous terms in the sequence. Instead of finding each term independently, a recursive formula builds each new term by applying a specific rule to one or more of its preceding terms. Here's how it works for our sequence:

• We start with an initial term, in this case, \(a_{1} = 3\).
• Then, for every subsequent term (like \(a_{n}\)), the formula \(a_{n} = n + 2a_{n-1}\) uses the previous term \(a_{n-1}\) to calculate the next one.

This makes recursive sequences a bit like a chain reaction, where knowing one part (the initial term) helps you find the next link and so on. Recursive formulas are powerful in various fields like computer science and mathematics as they simplify complex systems by breaking them down into basic, repeatable steps.
Whenever you come across a recursive sequence, identifying the initial term is crucial since it serves as the seed from which the rest of the sequence grows. Then, carefully apply the defined rule to progress through the series.

Sequence Progression

Sequence progression refers to the development and pattern emerging from the sequence as you compute more terms. In a recursive sequence, each term is linked to and depends on its predecessors, leading to a unique progression:

• Our sequence starts simply with \(a_{1} = 3\).
• The next term, \(a_{2}\), is calculated using \(a_{1}\), resulting in it being larger due to the formula and the recursive nature, giving us \(a_{2} = 8\).
• This pattern continues with \(a_{3} = 19\), \(a_{4} = 42\), and finally \(a_{5} = 89\).

Notice how each succeeding term grows larger, exhibiting a clear upward trend. The recursive rule makes every term depend on the previous, and this dependency shapes the progression's overall pattern.
Sequence progression as seen here becomes intuitive once you grasp the logic behind recursion. As you calculate more terms, you begin to anticipate the size and nature of future terms, which reflects the predictable nature of well-structured recursive sequences.

Mathematical Induction

Mathematical induction is a powerful technique used in proving propositions about sequences, particularly those defined recursively. It works similarly to falling dominoes — if you can prove it for the first term and show it holds from one term to the next, it must be true for all subsequent terms.Here's how it relates to our sequence:

• Base Case: Verify the proposition for \(a_{1} = 3\). For any statement about this sequence, you'd show it holds for this initial term.
• Inductive Step: Assume the proposition holds for some arbitrary term \(a_{k}\). Then prove it must also hold for the next term \(a_{k+1} = k+1 + 2a_{k}\).

By using these steps, induction gives us a robust tool to verify properties across the entire span of the sequence.
Whether it's confirming the form of the recursive formula or some other property, induction provides a solid foundation on which to build these proofs. It bridges the logic from a single term to an entire infinite set.