Problem 29 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 0.3,-0.09,0.027,-0.0081, \dots $$

Step-by-Step Solution

Answer

Common ratio is -0.3, fifth term is 0.00243, and nth term is \( 0.3 \times (-0.3)^{n-1} \).

Step 1: Identify the Common Ratio

To find the common ratio, divide the second term by the first term. For this sequence, calculate \( r = \frac{-0.09}{0.3} = -0.3 \). This common ratio confirms the sequence is geometric.

Step 2: Calculate the Fifth Term

Use the formula for the nth term of a geometric sequence, \( a_n = a_1 \times r^{n-1} \), where \( a_1 = 0.3 \) and \( r = -0.3 \). Substitute to find the fifth term: \( a_5 = 0.3 \times (-0.3)^{4} = 0.3 \times 0.0081 = 0.00243 \).

Step 3: Derive the nth Term Expression

The general formula for the nth term of a geometric sequence is \( a_n = a_1 \times r^{n-1} \). In this case, substitute \( a_1 = 0.3 \) and \( r = -0.3 \), resulting in \( a_n = 0.3 \times (-0.3)^{n-1} \).

Key Concepts

Common Ratio

The common ratio is a key concept in understanding geometric sequences. It is essentially the factor by which we multiply one term to get the next term in the sequence. To find the common ratio, you simply divide any term by the preceding term in the sequence.

In the given sequence, which starts with 0.3, and continues with -0.09, the common ratio can be determined as follows:

• Take the second term, -0.09.
• Divide it by the first term, 0.3: \( r = \frac{-0.09}{0.3} = -0.3 \)

This calculation shows that the common ratio is \(-0.3\). Once you have this ratio, you can confidently establish that the sequence is geometric, meaning each term is derived by multiplying the previous term by this common ratio.

Nth Term of a Geometric Sequence

The nth term of a geometric sequence allows you to find any term in the sequence without listing them all. This is incredibly useful in large sequences.

The formula to calculate it is:

• \( a_n = a_1 \times r^{n-1} \)

Here, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you want to find. For the sequence given, the first term \( a_1 \) is 0.3 and the common ratio \( r \) is -0.3. To find any term \( a_n \), substitute these values into the formula:

• \( a_n = 0.3 \times (-0.3)^{n-1} \)

This expression gives you the flexibility to find any nth term directly, which is powerful when dealing with very large sequences.

Fifth Term

To find the fifth term of a geometric sequence, you use the nth term formula, knowing the term number. The fifth term corresponds to \( a_5 \), where \( n = 5 \). Using the formula:

• \( a_5 = a_1 \times r^{4} \)

Substitute the values for this sequence:

• \( a_1 = 0.3 \)
• \( r = -0.3 \)

Substituting these into the formula:

• \( a_5 = 0.3 \times (-0.3)^4 = 0.3 \times 0.0081 \)
• This simplifies to: \( a_5 = 0.00243 \)

Therefore, the fifth term in the sequence is 0.00243. This calculation demonstrates how each subsequent term's position can dramatically change its value, especially as it moves through steps of multiplication by the common ratio.

Formula for Geometric Sequences

The formula for geometric sequences is essential for calculating terms and understanding the pattern of the sequence.

This formula is expressed as:

• \( a_n = a_1 \times r^{n-1} \)

This formula serves several purposes:

• Allows easy calculation of any term in the sequence.
• Facilitates the understanding of how each term is related to the one before it.
• Can be rearranged to find the common ratio if terms are known.

For the sequence in question, where the first term \( a_1 = 0.3 \) and the common ratio \( r = -0.3 \), the formula becomes:

• \( a_n = 0.3 \times (-0.3)^{n-1} \)

This personalized formula offers a clear path to uncovering the value of any term you need within this sequence. As each sequence may differ by starting term and ratio, the formula provides a universal approach to organizing and simplifying these patterns.