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Chapter 6: Problem 15

Find the first three terms of each recursively defined sequence. $$a_{1}=5, a_{n}=2 a_{n-1}$$

Short Answer

Expert verified
The first three terms of the recursively defined sequence are 5, 10, and 20.

Step by step solution

01

Identify the First Term

The first term \(a_1\) is already provided in the question, which is 5.
02

Find the Second Term

Apply the recursive rule to find the second term: \(a_2=2a_{1}\). Substitute \(a_{1}=5\) into this formula to get \(a_2=2*5=10\). So, the second term \(a_2\) is 10.
03

Find the Third Term

Similarly, now apply the recursive rule to find the third term: \(a_3=2a_{2}\). Substitute \(a_{2}=10\) into this formula to get \(a_3=2*10=20\). So, the third term \(a_3\) is 20.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Term
A recursively defined sequence starts with an initial value called the first term. This value is essential because it sets the stage for determining all subsequent terms in the sequence. In the given example, the first term is clearly stated as 5.

The first term is often denoted as \(a_1\). It provides the base case for the recursive formula that follows. Without a clear starting point, the sequence can't progress.

Remember:
  • The first term is fixed and isn't affected by the recursive process.
  • You always need to know this term before you can use the recursion rule to find the following terms.
Second Term
To find the second term in a recursively defined sequence, you apply the recursive rule using the first term as your base.

In our case, the rule is \(a_n = 2a_{n-1}\). So, to find \(a_2\), substitute \(a_1 = 5\) into the equation:\[ a_2 = 2a_1 = 2 \times 5 = 10 \]

This calculation illustrates how each term is derived from the preceding one. The process reveals the 'building block' nature of recursive sequences, where each step depends on the previous one.

Important points about second terms:
  • The second term is the first test of the recursive rule.
  • It confirms that the pattern described by the recursion is functioning as intended.
Third Term
Finding the third term involves applying the recursive formula again, this time using the second term.

Following our example, substitute \(a_2 = 10\) into the recursive equation:\[ a_3 = 2a_2 = 2 \times 10 = 20 \]

This completes the sequence of calculations needed to determine the first three terms. Notice how the sequence expands: each term requires the result of the previous computation.

Key takeaways for the third term:
  • The third term demonstrates the recursive relationship over more than one cycle.
  • It provides insight into the growth or change pattern dictated by the recursive formula.

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