Discovering the Pattern and Next Term in the Fraction Sequence -1/2 -1/4 -1/8

When faced with a sequence of fractions like -frac{1}{2} -frac{1}{4} -frac{1}{8}, it's often helpful to identify the underlying pattern. Understanding a sequence's pattern allows us to predict its subsequent terms and ensure a thorough comprehension of the mathematical relationship governing the sequence. In this article, we will delve into the pattern of the sequence -frac{1}{2} -frac{1}{4} -frac{1}{8} and determine the next term in the sequence.

Understanding the Fraction Sequence

The given sequence is: -frac{1}{2} -frac{1}{4} -frac{1}{8}. To identify the pattern, we should look at how each term is derived from the previous one. This process involves understanding the mathematical operations that transform one term into the next.

Determining the Pattern through Division

The sequence follows a clear pattern: each term is half of the previous term. This pattern can be easily identified by dividing each term by 2.

Step-by-Step Pattern Analysis

First term: -frac{1}{2} Second term: -frac{1}{4} -frac{1}{2} div 2 Third term: -frac{1}{8} -frac{1}{4} div 2 -frac{1}{2} div 2 div 2 -frac{1}{2} times frac{1}{4}

From the above, we observe that each term is the result of dividing the previous term by 2, or equivalently, multiplying the previous term by (frac{1}{2}).

Determining the Next Term

To find the next term in the sequence, we continue the pattern by dividing the last term by 2.

Calculation of the Next Term

Next term: -frac{1}{16} -frac{1}{8} div 2 -frac{1}{8} times frac{1}{2}

By recognizing the pattern, we can confidently state that the next term in the sequence is -frac{1}{16}.

Confirmation of the Pattern

We can further confirm the pattern by continuing the sequence:

Fourth term: -frac{1}{16} -frac{1}{8} div 2 -frac{1}{8} times frac{1}{2} Fifth term: -frac{1}{32} -frac{1}{16} div 2 -frac{1}{16} times frac{1}{2}

Thus, the complete sequence is: -frac{1}{2} -frac{1}{4} -frac{1}{8} -frac{1}{16} -frac{1}{32}

Understanding the Geometric Sequence

The sequence -frac{1}{2}, -frac{1}{4}, -frac{1}{8} is an example of a geometric sequence, where each term is obtained by multiplying the previous term by a constant ratio. In this case, the common ratio is (frac{1}{2}).

Geometric Ratio

The general form of a geometric sequence is a, ar, ar^2, ar^3, ..., where a is the first term and r is the common ratio. In our sequence:

First term (a): -frac{1}{2} Common ratio (r): frac{1}{2}

Using the common ratio, we can express the sequence as:

First term: a -frac{1}{2} Second term: ar -frac{1}{2} times frac{1}{2} -frac{1}{4} Third term: ar^2 -frac{1}{4} times frac{1}{2} -frac{1}{8} Fourth term: ar^3 -frac{1}{8} times frac{1}{2} -frac{1}{16}

Generalizing the Pattern

Given the geometric sequence and the pattern, we can express the nth term of the sequence using the formula for the nth term of a geometric sequence:

a_n a cdot r^{(n-1)}

Substituting the values, we get:

-frac{1}{2} cdot (frac{1}{2})^{(n-1)}

For the next term (i.e., when n4):

a_4 -frac{1}{2} cdot (frac{1}{2})^{3} -frac{1}{2} cdot frac{1}{8} -frac{1}{16}

Conclusion

Understanding the pattern and the underlying mathematical operations is crucial to identifying sequences and predicting their terms. In the sequence -frac{1}{2}, -frac{1}{4}, -frac{1}{8}, each term is half of the previous term, indicating a common ratio of (frac{1}{2}). By continuing this pattern, we can confidently state that the next term in the sequence is -frac{1}{16}.