Question

Find the next three terms of each sequence. (Lesson 1-1) $$5,10,20,40, \ldots$$

Short answer

The next three terms are 80, 160, and 320.

Step-by-step solution

Identify the pattern

We start by looking at the differences or ratios between the terms given in the sequence. In the sequence 5, 10, 20, 40, ..., each term is obtained by multiplying the previous term by 2.

Calculate the next term

To find the next term, take the last term given (40) and multiply it by the common ratio identified in Step 1, which is 2. Thus, the next term is \(40 \times 2 = 80\).

Calculate the second next term

To find the second next term, take the new term (80) and multiply it by 2. Thus, the second next term is \(80 \times 2 = 160\).

Calculate the third next term

Finally, take the second new term (160) and multiply it by 2 to find the third next term. Thus, the third next term is \(160 \times 2 = 320\).

Key concepts

Geometric progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This common ratio remains constant throughout the entire sequence, making it easy to predict subsequent terms.

• For example, consider the sequence given in the exercise: 5, 10, 20, 40. Here, each number is obtained by multiplying the previous number by 2, which is the common ratio.
• Geometric progressions are common in mathematics and appear in various real-world scenarios, such as calculating compound interest or population growth.
• Understanding geometric progressions can help to quickly identify the pattern and calculate future terms in the sequence.

Recognizing a geometric progression allows you to use its properties to your advantage. You can quickly determine any term in the sequence with the formula \[ a_n = a_1 imes r^{(n-1)} \] where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.

Common ratio

The common ratio is a key concept in understanding geometric progressions. It indicates how each term in the sequence is derived from the previous one. In the sequence given in the exercise,

• The first term is 5; to get to the second term, multiply 5 by the common ratio, which we have found to be 2, giving us 10.
• The common ratio (2) is the same for each pair of consecutive terms: 10 to 20 and 20 to 40, all follow the same pattern.
• Once the common ratio is known, predicting subsequent terms becomes straightforward. Simply keep multiplying the most recent term by the common ratio to find the next term.

A consistent common ratio is what defines the sequence as geometric. Determining this ratio is your first step when working with geometric sequences, and it helps simplify the process of finding any term in the sequence.

Identifying patterns

Identifying patterns in sequences is a fundamental skill in mathematics. Patterns help to recognize relationships between numbers and to predict future numbers in a sequence without directly calculating each one.

• In the exercise's sequence, the pattern involves multiplying by 2. Recognizing this pattern allows us to quickly and accurately determine future numbers.
• For any sequence, begin by examining the differences or ratios between terms. This is the first step towards revealing the inherent pattern.
• Understanding the type of sequence (geometric, arithmetic, etc.) can inform the strategy for finding future terms.

Once a pattern is identified, it can be applied to find the next few terms efficiently. This is especially useful in longer sequences where manually calculating each term could be tedious and error-prone. Recognizing patterns is thus a powerful tool for problem-solving in mathematical sequences.