Question

Give the next four terms in each sequence. \(64,32,16,8, \dots\)

Short answer

The next four terms are 4, 2, 1, 0.5.

Step-by-step solution

- Identify the Pattern

To find the next four terms in the sequence, first identify the pattern or rule governing the sequence. Observe the given terms: 64, 32, 16, 8. Notice that each term is obtained by dividing the previous term by 2.

- Apply the Pattern to Find the Next Term

Divide the last term given (which is 8) by 2 to obtain the next term. The next term is: 8 ÷ 2 = 4.

- Find the Second Next Term

Using the same pattern, divide 4 (the term found in Step 2) by 2. The next term is: 4 ÷ 2 = 2.

- Find the Third Next Term

Continue by dividing 2 (the term found in Step 3) by 2. The next term is: 2 ÷ 2 = 1.

- Find the Fourth Next Term

Lastly, divide 1 (the term found in Step 4) by 2. The next term is: 1 ÷ 2 = 0.5.

Key concepts

arithmetic progression

While the sequence provided in the exercise is not an example of an arithmetic progression, it's helpful to understand what arithmetic progressions are for broader learning. In arithmetic progression, each term is obtained by adding a fixed number, known as the common difference, to the previous term. For example, in the sequence 3, 6, 9, 12, ..., the common difference is 3 because each term is 3 more than the previous term. Arithmetic progressions are straightforward to identify and calculate, as you only need to keep adding the common difference to the last term you have.

division pattern

In the provided exercise, the main pattern involves division. Specifically, each term is obtained by dividing the previous term by 2. This type of sequence is known as a geometric sequence. The key to identifying a division pattern is to check for a consistent ratio between consecutive terms. If you notice that dividing any term by its preceding term gives you the same quotient, then you have identified a division pattern. In our example, 64 ÷ 32 = 2, 32 ÷ 16 = 2, 16 ÷ 8 = 2, etc. Understanding this division pattern helps simplify the process of finding subsequent terms by repeated division.

term identification

Term identification in sequences involves recognizing specific rules the sequence follows and applying them to find unknown terms. For the sequence 64, 32, 16, 8, ..., you start by noticing the division pattern mentioned earlier. Then, you proceed to identify the following terms methodically. First, divide 8 by 2 to get 4, then divide 4 by 2 to get 2, next divide 2 by 2 to get 1, and finally, divide 1 by 2 to get 0.5. Each of these steps involves simple arithmetic but must be followed precisely to ensure accuracy.

mathematical sequences

Mathematical sequences are ordered lists of numbers governed by specific rules or patterns. These rules help in predicting the next terms in the sequence. Common types of sequences include arithmetic progressions where you add a constant value, and geometric sequences where you multiply or divide by a constant value, as seen in our example. Being familiar with these types allows you to quickly recognize and work with sequences encountered in various mathematical contexts. Such clarity in understanding gives you the tools you need to solve sequence-related problems with confidence and accuracy.