Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{2}=-6, a_{7}=-192$$

Step-by-Step Solution

Answer

The first term is \(a_1 = -3\) and the common ratio is \(r = 2\).

Step 1: Understanding the Geometric Sequence Formula

In a geometric sequence, each term can be found using the formula \( a_n = a_1 \, r^{n-1} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.

Step 2: Expressing Given Terms with the Formula

We know from the problem \( a_2 = -6 \) and \( a_7 = -192 \). We can express these terms using the geometric sequence formula: 1. \( a_2 = a_1 r^{2-1} = a_1 r = -6 \) 2. \( a_7 = a_1 r^{7-1} = a_1 r^6 = -192 \)

Step 3: Solving for the Common Ratio (r)

By dividing the two equations, we eliminate \( a_1 \):\[ \frac{a_7}{a_2} = \frac{a_1 r^6}{a_1 r} = r^5 \]\[ \frac{-192}{-6} = r^5 \]\[ 32 = r^5 \] Taking the fifth root of both sides, we find:\[ r = 2 \]

Step 4: Solving for the First Term (a1)

Substitute \( r = 2 \) back into the equation \( a_1 r = -6 \):\[ a_1 \times 2 = -6 \]\[ a_1 = \frac{-6}{2} \]\[ a_1 = -3 \]

Step 5: Verifying the Solution

Use the found values of \( a_1 = -3 \) and \( r = 2 \) to check both terms:- For \( a_2 \): \[ a_2 = a_1 \times r = -3 \times 2 = -6 \]- For \( a_7 \): \[ a_7 = a_1 \times r^6 = -3 \times 2^6 = -3 \times 64 = -192 \]Both terms match the problem data, thus the answer is verified.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio (r) is the constant factor by which each term of the sequence is multiplied to get the next term. This is a key characteristic of a geometric sequence, distinguishing it from other sequences like arithmetic sequences, where a constant is added to each term instead.
Understanding the common ratio allows us to find any term in the sequence as long as one term and the ratio are known. We determine the common ratio by dividing a term by its preceding term.

• For example, if the second term (a_2) is -6, and the difference between indices is 5, leading to the seventh term (a_7) being -192, we use the ratio formula:\[ \frac{a_7}{a_2} = \frac{-192}{-6} = 32 = r^5 \]
• Taking the fifth root of 32 gives us the value of the common ratio: \( r = 2 \).

Thus, the common ratio is crucial in confirming the geometric nature of a sequence and developing its general formula.

Geometric Formula

The geometric formula is an essential tool used to identify and compute terms in a geometric sequence. The standard form is \( a_n = a_1 \, r^{n-1} \), where:

• \( a_n \) is the nth term of the sequence,
• \( a_1 \) is the first term, and
• \( r \) is the common ratio.

This formula allows us to find any term in the sequence without listing all previous terms.
For example, if you know terms \( a_2 = -6 \) and \( a_7 = -192 \), and find the common ratio \( r = 2 \), substituting back gives us:

• (For \( a_2 \)):\[ a_2 = a_1 \times r = -3 \times 2 = -6 \]
• (For \( a_7 \)):\[ a_7 = a_1 \times r^6 = -3 \times 2^6 = -192 \]

These calculations confirm the formula’s accuracy in linking the terms of the sequence correctly.

Sequence Verification

Sequence verification confirms the correctness of the identified sequence terms by substituting the derived values back into the formula. It involves critical checks that ensure solutions are accurate and equations hold true for given conditions.
To verify, utilize the values of the first term \( a_1 \) and the common ratio \( r \) found from other terms.

• Check \( a_2 \): With \( a_1 = -3 \) and \( r = 2 \), calculate: \[ a_2 = a_1 \times r = -3 \times 2 = -6 \]
• Check \( a_7 \): With the same values, calculate: \[ a_7 = a_1 \times r^6 = -3 \times 2^6 = -192 \]

These checks reinforce confidence in the solution's validity, showing that the terms correctly satisfy the geometric sequence equation.