Question

Determine the \(n^{\text {th }}\) term of the given sequence. $$10,20,40,80,160, \ldots$$

Short answer

The nth term is \( a_n = 10 \cdot 2^{n-1} \).

Step-by-step solution

Identify the Pattern

Look at the sequence of numbers: 10, 20, 40, 80, 160, .... Each number seems to be double the previous one. This indicates a geometric sequence with a common ratio.

Calculate the Common Ratio

To find the common ratio (r), divide the second term by the first: \( \frac{20}{10} = 2 \). Similarly, check other terms: \( \frac{40}{20}, \frac{80}{40}, \frac{160}{80} \), which also equals 2. Hence, the common ratio is 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 \( a_1 = 10 \) is the first term and \( r = 2 \) is the common ratio.

Express the nth Term

Plug the values \( a_1 = 10 \) and \( r = 2 \) into the formula: \[ a_n = 10 \cdot 2^{n-1} \]. This expression gives the nth term of the geometric sequence.

Key concepts

Common Ratio

The common ratio in a geometric sequence is a key element. It's what you multiply each term by to get to the next term in the sequence. To find it, take any term in the sequence, and divide it by the previous term. In our exercise, the sequence begins with 10, 20, 40, 80, 160. By dividing 20 by 10, we find the common ratio is 2. This same process applies to other consecutive terms:

• 40 divided by 20
• 80 divided by 40
• 160 divided by 80

All result in 2. This implies consistent multiplication throughout the sequence. Understanding the common ratio confirms the multiplicative pattern that defines geometric sequences.

Nth Term Formula

The nth term formula for a geometric sequence helps us find any term in the sequence without listing all the terms. Once you've got the common ratio, the formula is quite straightforward: \[ a_n = a_1 \cdot r^{n-1} \] Here, \( a_n \) represents the nth term. For the first term, \( a_1 \), and the common ratio, \( r \), you insert those values. In our example, the first term is 10, and \( r \) is 2. So the formula becomes:

• \[ a_n = 10 \cdot 2^{n-1} \]

This provides a way to calculate any term directly by plugging in the value for \( n \). It makes working with large numbers easy!

Sequence Pattern

Understanding the pattern in a sequence is crucial. In a geometric sequence, the pattern is defined by the common ratio. Once you've noticed that each term is obtained by multiplying the previous term by a constant number, you've identified a geometric sequence. In our example:

• Start with 10, multiply by 2 to get 20.
• Multiply 20 by 2 to get 40.
• This continuous pattern demonstrates the power of sequences.

Recognizing such patterns aids in easily predicting any following terms without needing to compute each one step by step. This helps streamline calculations and deepen the understanding of mathematical sequences.