Problem 2 - Algebra for College Students

Question

Find the fifth term of the sequence for which \(a_{n}=3(2)^{n-1}, \quad 48\)

Step-by-Step Solution

Answer

The fifth term is 48.

Step 1: Understanding the Problem

We are given a sequence defined by the formula \(a_{n}=3(2)^{n-1}\). We need to find the fifth term of this sequence. The formula means for any term \(a_n\), we substitute \(n\) to find the term value.

Step 2: Substitute to Find the Fifth Term

To find the fifth term \(a_5\), substitute \(n=5\) into the sequence formula: \(a_{5}=3(2)^{5-1}\). Simplify the exponent first.

Step 3: Calculate the Exponent

Calculate \(2^{5-1}\) which is \(2^4\). This results in \(2^4 = 16\). Now, we replace this value back into the expression for \(a_5\).

Step 4: Substitute the Exponent Result

Using the result from the previous step, substitute \(16\) into \(a_5 = 3 imes 16\).

Step 5: Multiply to Get the Fifth Term

Compute \(3 \times 16 = 48\). Hence, the fifth term of the sequence is 48.

Key Concepts

Sequence Formula

The sequence formula is a tool that helps us find specific terms in a sequence. For the given problem, the formula is \(a_{n}=3(2)^{n-1}\). This defines the entire sequence, allowing us to compute any term by simply substituting the position number, denoted by \(n\).
This sequence is a geometric sequence, defined by a constant ratio. In our example, the constant ratio is \(2\), which means each term is multiplied by \(2\) to get the next term.
The starting term (the first term, \(a_1\)) is produced by substituting \(n = 1\) in the formula, resulting in \(3(2)^{1-1} = 3\). Each subsequent term is found similarly.

• First Term: For \(n = 1\), the formula becomes \(3(2)^{1-1} = 3\).
• Fifth Term: For \(n = 5\), it's \(3(2)^{5-1}\).

Using a sequence formula helps in understanding patterns and calculating specific terms efficiently.

Exponentiation

Exponentiation is a fundamental concept in mathematics that involves raising a number (the base) to a power (the exponent). In the sequence formula \(a_{n}=3(2)^{n-1}\), exponentiation is key to determining the correct term value.
Here, the number \(2\) is raised to the power of \(n-1\). This means you multiply \(2\) by itself \(n-1\) times. For instance, to find the fifth term, the power is \(5-1=4\), leading to \(2^4\).

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)

Understanding exponentiation greatly aids in sequences involving growth or decay, making it easier to calculate how quickly terms in a geometric sequence increase.

Term Calculation

Term calculation involves evaluating the sequence formula for a specific \(n\) to determine the value of that term. For the sequence \(a_{n}=3(2)^{n-1}\), calculating terms is straightforward. We substitute the position \(n\) into the formula and compute the result step-by-step.
To find the fifth term, substitute \(n = 5\) into the formula, giving us \(3(2)^{5-1}\) as the expression. Calculate the exponent first, \((2^4 = 16)\), then multiply by 3.
This gives us:

• Calculate the exponent: \(2^{5-1} = 2^4 = 16\)
• Multiply: \(3 \times 16 = 48\)

Following these steps ensures accurate term calculation, showing how each calculation builds on the previous steps. Term calculation is the final step to finding specific terms in a sequence, highlighting the importance of substituting and computing precisely.