Problem 48 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{1}=4096, r=\frac{1}{4}, n=8 $$

Step-by-Step Solution

Answer

The 8th term is \(\frac{1}{4}\).

Step 1: Identify the Formula

To find a specific term in a geometric sequence, use the formula for the nth term: \[a_n = a_1 \cdot r^{(n-1)}\]where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Input Known Values

We know from the problem that \(a_1 = 4096\), \(r = \frac{1}{4}\), and \(n = 8\). Substitute these values into the formula:\[a_8 = 4096 \cdot \left(\frac{1}{4}\right)^{(8-1)}\]

Step 3: Calculate the Common Ratio to the Power of (n-1)

First, compute \(r^{(n-1)}\):\[\left(\frac{1}{4}\right)^{7} = \frac{1}{16384}\]

Step 4: Compute the 8th Term

Substitute the value from step 3 back into the formula:\[a_8 = 4096 \cdot \frac{1}{16384} = \frac{4096}{16384} = \frac{1}{4}\]Therefore, the 8th term of the sequence is \(\frac{1}{4}\).

Key Concepts

Nth Term of a Geometric Sequence

The nth term in a geometric sequence is a specific term located at position "n" in the sequence. To find it, you need to know the first term and the common ratio. The formula for the nth term is: \[a_n = a_1 \cdot r^{(n-1)}\] In this formula, \(a_n\) represents the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) indicates the position of the term in the sequence.

• If \(n = 1\), then \(a_n\) is simply \(a_1\) since you don't have to multiply by the common ratio yet.
• As \(n\) increases, \(r^{(n-1)}\) plays a major role because it exponentially changes the amount you're multiplying \(a_1\) by.

Understanding how to use this formula is crucial for finding any term in a geometric sequence, no matter how large \(n\) is.

Understanding the Common Ratio

The common ratio in a geometric sequence is the consistent factor by which you multiply each term to get to the next one. In simpler terms, it is the number that determines how the sequence progresses or changes.

• If the common ratio \(r\) is greater than 1, the sequence multiplies, resulting in a growing pattern.
• If \(0 < r < 1\), the sequence shrinks, as in the example given with \(r = \frac{1}{4}\), which gradually reduces the size of each term.
• If \(r\) is negative, the sequence will alternate in sign, switching between positive and negative terms.

Understanding the common ratio is key because it dictates the behavior and pattern of the sequence. Knowing \(r\) allows you to predict how the sequence continues or changes.

Geometric Sequences Formulas

Geometric sequences have a powerful set of formulas that allow you to calculate various aspects of the sequence quickly and accurately:

• Nth Term Formula: This is the core formula used to find any term in a sequence: \[a_n = a_1 \cdot r^{(n-1)}\]
• Sum of a Finite Geometric Series: This formula computes the sum of the first \(n\) terms: \[S_n = a_1 \cdot \frac{1-r^n}{1-r}\]
• Sum of an Infinite Geometric Series: When \(|r| < 1\), you can use this formula: \[S = \frac{a_1}{1-r}\]

These formulas are incredibly useful for uncovering characteristics of sequences and solving complex mathematical problems. Each has its own application and purpose, giving you tools to handle different situations efficiently.

Using Exponents in Sequences

Exponents are foundational in understanding and working with geometric sequences. In the nth term formula, exponents are used to elevate the common ratio to the power of \(n-1\). This exponentiation has significant effects:

• Growth or Decay: Depending on \(r\) (greater than 1 means growth; less than 1 signals decay), raising \(r\) to higher powers affects the rate at which the sequence expands or contracts.
• Calculations: Computing \(r^{(n-1)}\) is essential because it translates the linear form of addition into the exponential form needed for multiplication to achieve geometric growth.
• Predictive Power: Exponents allow mathematicians to model complex phenomena where growth or reduction happens multiply rather than additively.

Understanding how exponents function in these sequences helps in predicting the behavior and outcome of the series accurately, making them an indispensable mathematical tool.