Problem 44 - Precalculus: Mathematics for Calculus

Question

Find the indicated term(s) of the geometric sequence with the given description. The third term is \(-54\) and the sixth term is \(\frac{729}{256} .\) Find the first and second terms.

Step-by-Step Solution

Answer

First term: -24, Second term: 36.

Step 1: Understanding the Problem

We are given that the third term of a geometric sequence is \(-54\) and the sixth term is \(\frac{729}{256}\). We need to find the first and second terms of this sequence.

Step 2: Identify the Formula

The general term of a geometric sequence can be writing as \(a_n = a_1 \, r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 3: Set Up the Equations

From the problem statement, the third term \(a_3 = a_1 \, r^2 = -54\) and the sixth term \(a_6 = a_1 \, r^5 = \frac{729}{256}\). We have two equations: \(a_1 \, r^2 = -54\) and \(a_1 \, r^5 = \frac{729}{256}\).

Step 4: Solve for Common Ratio

Divide the equation for the sixth term by the equation for the third term to solve for \(r^3\): \(\frac{a_1 \, r^5}{a_1 \, r^2} = \frac{\frac{729}{256}}{-54}\). Simplifying gives \(r^3 = -\frac{27}{16}\).

Step 5: Calculate the Common Ratio

Take the cube root of both sides to find \(r\): \(r = -\left(\frac{27}{16}\right)^{1/3} = -\frac{3}{2}\).

Step 6: Find the First Term

Substitute \(r = -\frac{3}{2}\) into the third term equation: \(a_1 \left(-\frac{3}{2}\right)^2 = -54\). Solving for \(a_1\) gives \(a_1 \cdot \frac{9}{4} = -54\). Hence, \(a_1 = -54 \times \frac{4}{9} = -24\).

Step 7: Find the Second Term

Using \(a_1 = -24\) and the common ratio \(r = -\frac{3}{2}\), the second term \(a_2 = a_1 \, r = -24 \, \times \left(-\frac{3}{2}\right) = 36\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element used to determine how the sequence progresses. Imagine it as the factor by which each term is multiplied to get the next term. Understanding the common ratio, denoted as \( r \), allows us to predict subsequent terms effortlessly.

To find the common ratio in the given problem, we divided the sixth term equation by the third term equation—essentially comparing how much the sequence grows over the three-term span. Simplifying this gives us a neat formula:

• \( r^3 = -\frac{27}{16} \)

This equation lets us solve for \( r \) by taking the cube root. Remember, the common ratio can be positive or negative, impacting how the sequence behaves. A negative \( r \) results in alternating signs between terms, as seen in the exercise where \( r = -\frac{3}{2} \).

Therefore, identifying the common ratio is fundamental in unlocking all sequence values.

Geometric Progression Formula

The geometric progression formula is central to solving problems involving geometric sequences. It takes the form \( a_n = a_1 \, r^{n-1} \), where \( a_n \) is any term you wish to find, \( a_1 \) is the first term, and \( r \) is the common ratio.

This formula is essentially your cookbook recipe for geometric sequences. You insert known values like \( a_1 \) and \( r \), then solve for your desired term. In our task, we used this formula to set up equations based on the third and sixth terms:

• \( a_3 = a_1 \, r^2 = -54 \)
• \( a_6 = a_1 \, r^5 = \frac{729}{256} \)

These equations allowed us to systematically solve for unknowns step-by-step. Knowing this formula, you can tackle any term in the sequence once you know the first term and the common ratio. It's all about substituting values and simplifying.

Sequence Terms Calculation

Calculating sequence terms in a geometric sequence involves applying the geometric progression formula with precise substitutions. Without loss of generality, the goal is often to find earlier or intermediary terms from later ones using known sequence properties.

In this exercise, after determining the common ratio \( r = -\frac{3}{2} \) and using the given third term \(-54\), we start by recalculating the first term \(a_1\). The equation derives from the geometric progression formula for the third term:

• \( a_1 \, \left(-\frac{3}{2}\right)^2 = -54 \)

Solve for \( a_1 \) to establish the foundation:

• \( a_1 = -54 \times \frac{4}{9} = -24 \)

Armed with \( a_1 \), calculating the second term becomes straightforward:

• \( a_2 = a_1 \, r = -24 \, \times \left(-\frac{3}{2}\right) = 36 \)

In essence, this process is iterative application of known values and the formula. It makes solving for sequence terms feel more like solving a puzzle with logical steps and substitutions.