Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=6 ; \quad r=-2 $$

Short answer

The fifth term is 96, and the nth term is \(a_n = 6 \times (-2)^{(n-1)}\).

Step-by-step solution

Identify Given Values

The first term of the geometric sequence is given as \(a_1 = 6\), and the common ratio is given as \(r = -2\).

Use the Geometric Sequence Formula

The nth term of a geometric sequence can be found using the formula \(a_n = a_1 \times r^{(n-1)}\).

Find the Fifth Term

Substitute \(n = 5\), \(a_1 = 6\), and \(r = -2\) into the formula: \[ a_5 = 6 \times (-2)^{(5-1)} = 6 \times (-2)^4 \] Calculate \((-2)^4\): \[ (-2)^4 = 16 \] Then calculate \(a_5\): \[ a_5 = 6 \times 16 = 96 \] So, the fifth term is 96.

General Form of the nth Term

To find the general form for the nth term, substitute \(a_1\) and \(r\) into the nth term formula: \[ a_n = 6 \times (-2)^{(n-1)} \] So, the nth term is given by \(a_n = 6 \times (-2)^{(n-1)}\).

Key concepts

nth term formula

To understand a geometric sequence, knowing how to find the nth term is crucial. The nth term formula for a geometric sequence helps you find any term in the sequence easily. The formula is given by \( a_n = a_1 \times r^{(n-1)} \). Here, \( a_n \) represents the nth term, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the term number you want to find.

Using this formula, you can calculate any term without having to list all the previous terms, making it very efficient for large sequences or higher values of \( n \).

For our example:

• First term: \( a_1 = 6 \)
• Common ratio: \( r = -2 \)

Plugging these values into the nth term formula gives:
\( a_n = 6 \times (-2)^{(n-1)} \).

common ratio

The common ratio \( r \) in a geometric sequence is the factor by which each term is multiplied to get the next term. It remains constant throughout the sequence.

We find the common ratio by dividing any term by the previous term. In our problem, the first term is 6 and the common ratio is given as -2. This means each term is obtained by multiplying the previous term by -2.

Understanding the common ratio helps in:

• Predicting the nature (positive or negative) of subsequent terms
• Determining the growth factor of the sequence
• Providing insights into exponential growth or decay in mathematical and real-world scenarios

In this example, the sequence alternates in sign because the common ratio \( r = -2 \) is negative.

fifth term calculation

To calculate the fifth term in our geometric sequence, we use the nth term formula and substitute \( n = 5 \).

The given values are:

• First term \( a_1 = 6 \)
• Common ratio \( r = -2 \)

Substitute these values into the formula \( a_n = a_1 \times r^{(n-1)} \):

\[ a_5 = 6 \times (-2)^{(5-1)} \]
First, evaluate the exponent:

\[ (-2)^4 = 16 \]
Next, multiply by the first term:

\[ a_5 = 6 \times 16 = 96 \]
Therefore, the fifth term of the sequence is 96.

general term of sequence

The general term of a geometric sequence provides a way to find any term in the sequence without having to list all previous terms. It's derived from the nth term formula.

To find the general term, substitute \( a_1 \) and \( r \) into the nth term formula:

In the given problem:

• \( a_1 = 6 \)
• \( r = -2 \)

Therefore, the general term is:
\[ a_n = 6 \times (-2)^{(n-1)} \]
This means for any term number \( n \), you can find the term by plugging \( n \) into this formula. Whether you need the 5th, 10th, or 100th term, this formula will quickly provide the correct term in the sequence.