Problem 27 - Beginning and Intermediate Algebra

Question

Find the general term of each geometric sequence. $$3,-6,12,-24, \dots$$

Step-by-Step Solution

Answer

The general term \(T_n\) of the geometric sequence \(3,-6,12,-24,\dots\) is: \[T_n = 3(-2)^{n-1}\]

Step 1: Identify the first term#a_1# and the common ratio#r

Given the geometric sequence: $$3,-6,12,-24,\dots$$ The first term \(a_1\) is 3. To find the common ratio (\(r\)), divide the second term by the first term, the third term by the second term, and so on. $$r = \frac{-6}{3} = -2$$

Step 2: Use the formula to find the general term

Now that we have the first term \(a_1\) and the common ratio \(r\), we can find the general term of the sequence by using the geometric sequence formula: \[T_n = a_1 r^{n-1}\] Now replace the \(a_1\) and \(r\) with their values: \[T_n = 3(-2)^{n-1}\] So, the general term \(T_n\) of the geometric sequence is: \[T_n = 3(-2)^{n-1}\]

Key Concepts

Common Ratio

In a geometric sequence, the common ratio (\( r \)) is a crucial component that helps define and determine each subsequent term in the sequence. It remains consistent throughout the sequence. To find the common ratio, you divide any term by the term immediately preceding it.
For the sequence given, \( 3, -6, 12, -24, \) ..., we calculate the common ratio as follows:

• Take the second term (-6) and divide by the first term (3): \( r = \frac{-6}{3} = -2 \).
• Similarly, verify with the next pair: \( \frac{12}{-6} = -2 \).

This consistent ratio of -2 dictates how each number in the sequence relates to the next.

General Term

The general term formula of a geometric sequence is a compact way that allows us to express any term within the sequence without listing out all preceding terms. The formula is:\[T_n = a_1 r^{n-1}\]where:

• \( T_n \) is the n-th term,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the term number.

For example, in the given sequence, the general term is:\[T_n = 3(-2)^{n-1}\]This equation enables quick calculation of any desired term. Just substitute the \( n \) value to find the associated term in the sequence. For instance, \( T_3 \) (third term) would be \( 3(-2)^{2} = 3 \times 4 = 12 \).

First Term

In any geometric sequence, the first term (\( a_1 \)) is a foundational part of determining the entire sequence. It is the starting point from which all other terms derive.
For the sequence \( 3, -6, 12, -24, \) ..., the first term is \( a_1 = 3 \).This first term is crucial because the general formula of a geometric sequence builds upon it: \[T_n = a_1 r^{n-1}\]Starting off with the correct first term ensures the integrity of term calculations all along the sequence. Thus, getting the value of \( a_1 \) right is essential for defining the precise path of the arithmetic progression.