Problem 15 - Algebra and Trigonometry

Question

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=3 \text { and } a_{n}=4 a_{n-1} \text { for } n \geq 2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 3, 12, 48, 192.

Step 1: Determine the First Term

The first term of the sequence, denoted as \(a_1\), is given in the problem statement. We can see that the first term of the sequence is 3. Thus, \(a_1 = 3\).

Step 2: Apply the Recursion Formula for the Second Term

Now we use the recursion formula given as \(a_n = 4a_{n-1}\) for \(n \geq 2\) to find the second term. Here, \(n = 2\). So, we substitute \(n = 2\) into the recursion formula to find that \(a_2 = 4a_{2-1} = 4a_1\). By substituting the value of \(a_1 = 3\) into this formula, we find that \(a_2 = 4 * 3 = 12\). Thus, the second term of the sequence is 12.

Step 3: Apply the Recursion Formula for the Third Term

Following the same process, we can find the third term by substituting \(n = 3\) into the recursion formula. We find that \(a_3 = 4a_{3-1}= 4a_2\). By substituting the value of \(a_2 = 12\) into this formula, we find that \(a_3 = 4 * 12 = 48\). Thus, the third term of the sequence is 48.

Step 4: Apply the Recursion Formula for the Fourth Term

Then, for the fourth term, we substitute \(n = 4\) into the recursion formula to get \(a_4 = 4a_{4-1}= 4a_3\). By substituting the value of \(a_3 = 48\) into this formula, we find that \(a_4 = 4 * 48 = 192\). Thus, the fourth term of the sequence is 192.

Key Concepts

Recursion Formulas

Recursion formulas are like mathematical instructions. They tell you how to get from one term in a sequence to the next. In simple terms, if you know one term, the recursion formula helps you find the next one. This is done by using a rule that involves the previous term(s).

• These formulas start with an initial term, often labeled as \(a_1\).
• Then, a relationship is given for \(a_n\), helping calculate subsequent terms using prior ones.

As an example, consider our sequence: we start with \(a_1 = 3\), and the given recursion formula is \(a_n = 4a_{n-1}\), which means each term is four times the previous term. By applying this formula, you can construct the sequence step by step.

Sequence Terms

Sequence terms are the individual elements in a sequence. They form a chain that follows a specific pattern or rule. In our problem, these terms are generated using the recursive formula we have discussed.

• The sequence begins with an initial term, often directly provided.
• Each subsequent term is calculated based on the previous term, using the recursion formula.
• The terms we've calculated are \(a_1 = 3\), \(a_2 = 12\), \(a_3 = 48\), and \(a_4 = 192\).

Understanding each term helps in comprehending the broader pattern of the sequence. It also reveals how recursive sequences expand or contract depending on the rule applied to the previous terms.

Algebraic Sequences

Algebraic sequences are sequences where their terms have a definite pattern often expressed using methods from algebra. Recursion is one such method.

• Using algebraic rules makes it easier to generate the terms of the sequence.
• This helps in predicting future terms without listing every previous term.
• In our scenario, the algebraic sequence is defined using \(a_n = 4a_{n-1}\). Each term grows by a factor of 4, showing a pattern of exponential growth.

Algebraic sequences, while at times complex, give a systematic way to explore relationships between numbers, making them applicable in various fields of mathematics and real-world problems.