Problem 26 - Beginning and Intermediate Algebra

Question

Find the general term of each geometric sequence. $$-1,4,-16,64, \dots$$

Step-by-Step Solution

Answer

The general term of the given geometric sequence is: \( a_n = -1 \cdot (-4)^{n-1} \)

Step 1: Determine the common ratio

To find the common ratio, divide the second term by the first term, the third term by the second term, and so on. $$\frac{4}{-1} = -4$$ $$\frac{-16}{4} = -4$$ $$\frac{64}{-16} = -4$$ Since the common ratio is the same for every pair of consecutive terms, the common ratio (r) is -4.

Step 2: Use the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is: $$ a_n = a_1 \cdot r^{n-1} $$ where - \(a_n\) is the nth term of the sequence, - \(a_1\) is the first term, - r is the common ratio, and - n is the term number. In our case, \(a_1 = -1\) and \(r = -4\).

Step 3: Substitute values and find the general term

Using the given values for \(a_1\) and r, we can find the general term of the geometric sequence: $$ a_n = -1 \cdot (-4)^{n-1} $$ So, the general term of the given geometric sequence is: $$ a_n = -1 \cdot (-4)^{n-1} $$

Key Concepts

Common Ratio

In a geometric sequence, understanding the common ratio is crucial. The common ratio is a constant value that you obtain by dividing any term in the sequence by its preceding term. This ratio allows the sequence to be consistent across all terms. For instance, in the sequence given as

• -1, 4, -16, 64,

you find the common ratio by calculating

• \( \frac{4}{-1} = -4 \)
• \( \frac{-16}{4} = -4 \)
• \( \frac{64}{-16} = -4 \)

These calculations confirm the common ratio \(r\) is \(-4\). With this ratio, each term can be deduced from the previous one, maintaining a uniform growth pattern. Recognizing and verifying the common ratio ensures you can accurately describe and analyze the sequence.

nth Term Formula

A pivotal aspect of working with geometric sequences is using the nth term formula. This formula helps you calculate any term in the sequence without listing every term up to that point. The nth term formula for a geometric sequence is given as

• \( a_n = a_1 \cdot r^{n-1} \)

where

• \( a_n \) represents the term you want to find,
• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio, and
• \( n \) is the term number you are interested in.

For instance, with the sequence starting at \(-1\) and having a ratio of \(-4\), the formula becomes

• \( a_n = -1 \times (-4)^{n-1} \).

This equation simplifies finding any term by just plugging in the desired term number \(n\), making the process systematic and efficient.

General Term Calculation

Calculating the general term of a geometric sequence makes it easier to identify any term within the sequence. By substituting known values for \( a_1 \) and \( r \) into the nth term formula, you derive the general term equation. From the original sequence, with \( a_1 = -1 \) and \( r = -4 \), the general term calculation becomes:

• \( a_n = -1 \cdot (-4)^{n-1} \).

This form succinctly expresses how each term is calculated, providing a quick reference for determining any term in the sequence structurally rather than iteratively. With the sequence now defined by its general term, you can effortlessly explore, analyze, and predict its progression, easing assignments and helping in deeper mathematical explorations.