Understanding Sequences with Fractions and Decimals

When dealing with sequences in mathematics, especially those involving fractions and decimals, it's important to recognize the pattern or rule that governs the progression of the sequence. In the sequence provided, 2, 2.5, 3, __, 4, we need to identify the pattern and determine the missing term.

Method 1: Identifying the Pattern Through Differences

One way to solve the sequence is to look at the differences between the terms.

Let's start by writing the terms in a more familiar decimal form:

2 2.5 (which is 5/2) 3 4

Between each consecutive term, we observe a difference of 0.5. For instance:

2.5 - 2 0.5 3 - 2.5 0.5

Following this pattern, the next term should be 3 0.5 3.5. Therefore, the next term in the sequence is 3.5, which can also be written as 7/2.

Method 2: Using Fractions to Identify the Pattern

Another method involves expressing the sequence in a common fraction form. This helps us recognize the underlying pattern more clearly.

Writing each term as a fraction with a common denominator (in this case, 2), we get:

4/2 (which is 2) 5/2 (which is 2.5) 6/2 (which is 3) __ (we need to find this term) 8/2 (which is 4)

From this, we can see that the numerator increases by 1 for each subsequent term. Hence, the missing term's numerator should be one more than the previous term, which is 7/2.

Conclusion

Both methods yield the same result: the missing term in the sequence is 7/2 or 3.5. This demonstrates how understanding patterns in fractions and decimals can help solve complex sequence problems.

Relevant Keywords

Fractions, Decimals, Sequence