Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Find the seventh term of the geometric sequence whose second and third terms are 2 and \(-\sqrt{2}\).

Step-by-Step Solution

Answer

The seventh term is \(-\frac{\sqrt{2}}{4}\).

Step 1: Identify the given terms

We are given the second term of the sequence as 2 and the third term as \(-\sqrt{2}\). We need to use these terms to find the common ratio \(r\) of the geometric sequence.

Step 2: Calculate the common ratio

In a geometric sequence, each term is the previous term multiplied by the common ratio \(r\). Thus, we have the equation for the third term: \[2 \times r = -\sqrt{2}\]Solving for \(r\), we divide both sides by 2:\[r = \frac{-\sqrt{2}}{2}\]

Step 3: Write the formula for the nth term

The nth term of a geometric sequence can be given by the formula:\[a_n = a_1 \times r^{n-1}\]To find the seventh term, we need the first term \(a_1\), which can be calculated knowing that the second term \(a_2 = 2\) is given by:\[a_2 = a_1 \times r = 2\]

Step 4: Calculate the first term

Using the expression for the second term:\[a_1 \times r = 2\]Replacing \(r\) with \(\frac{-\sqrt{2}}{2}\):\[a_1 \times \frac{-\sqrt{2}}{2} = 2\]Solving for \(a_1\), multiply both sides by \(-2/\sqrt{2}\):\[a_1 = \frac{-4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -2\sqrt{2}\]

Step 5: Calculate the seventh term

Now that we have \(a_1\) and \(r\), we can find the seventh term using \(n=7\) in the formula:\[a_7 = a_1 \times r^{6}\]Substitute \(a_1 = -2\sqrt{2}\) and \(r = -\frac{\sqrt{2}}{2}\):\[a_7 = (-2\sqrt{2}) \times \left(-\frac{\sqrt{2}}{2}\right)^6\]Calculate \(\left(-\frac{\sqrt{2}}{2}\right)^6\):\[\left(-\frac{\sqrt{2}}{2}\right)^6 = \left(\frac{2}{4}\right)^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}\]Thus, substituting back:\[a_7 = -2\sqrt{2} \times \frac{1}{8} = -\frac{\sqrt{2}}{4}\]

Step 6: Conclusion

The seventh term of the sequence is \(-\frac{\sqrt{2}}{4}\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element. It defines the relationship between consecutive terms in the sequence. To find the common ratio \(r\), you can simply divide any term by the preceding term. In the exercise, you are provided with the second term \(2\) and the third term \(-\sqrt{2}\).

• To get the common ratio: divide the third term by the second term.

This gives:\[r = \frac{-\sqrt{2}}{2}\]
The common ratio stays constant for the sequence, meaning every term is the previous term times \(r\). This uniformity is what defines a geometric sequence and differentiates it from arithmetic sequences which use a common difference instead.

Nth Term Formula

The nth term of a geometric sequence gives you a clear way to find any term in the sequence without listing all the previous ones. This formula is:

• \( a_n = a_1 \times r^{n-1} \)

Here, \(a_n\) is the term you want to find, \(a_1\) is the first term of the sequence, and \(r\) is the common ratio. The formula shows the power of exponential growth in geometric sequences; each term is a result of multiplying the first term by \(r\) raised to the power of \((n-1)\).
This allows you to jump straight to any term without the tedious work of calculating each sequentially, which can save time and effort, especially with larger sequences.

Seventh Term Calculation

In this exercise, finding the seventh term requires the nth term formula. You already have \(a_1\) and \(r\), so you substitute these into your formula to find the seventh term \(a_7\). Set \(n\) to 7, apply the values:

• \( a_7 = a_1 \times r^6 \)

In our problem:

• \( a_1 = -2\sqrt{2} \)
• \( r = -\frac{\sqrt{2}}{2} \)

We find:\[a_7 = (-2\sqrt{2}) \times \left(-\frac{\sqrt{2}}{2}\right)^6 = -\frac{\sqrt{2}}{4}\]Calculating the sixth power of \(r\) first serves you a clearer perspective of the influence of the ratio. The calculation reveals how components of the sequence evolve as you move to further terms.

First Term of the Sequence

The first term \(a_1\) is fundamental in a geometric sequence because all other terms build from this starting point. When you're given other terms in the sequence, you reverse-engineer the first term using the known term and the common ratio. In this sequence, you started with the second term plus the common ratio to find \(a_1\):

• Given: \( a_2 = a_1 \times r = 2 \)
• Insert \( r = \frac{-\sqrt{2}}{2} \)

Solve for \(a_1\) by multiplying both sides by the reciprocal:\[a_1 = \frac{-2}{\frac{-\sqrt{2}}{2}} = -2\sqrt{2}\]
This calculation corrects your course to ensure each subsequent term aligns with the foundational ratio transformation. Knowing \(a_1\) is always essential when using the general \(a_n\) formula.