Determine the formula for the general term of the following sequence: 1/2, 1, 3/2, 2, 5/2, ...
Understand the Problem
The question is asking to find a formula for the general term of a given sequence of fractions. The provided sequence starts with 1/2, 1, 3/2, 2, and 5/2, which suggests a pattern or formula must be derived from these terms.
Answer
The general term of the sequence is given by \( a_n = \frac{n}{2} \).
Answer for screen readers
The formula for the general term of the sequence is: $$ a_n = \frac{n}{2} $$
Steps to Solve
- Identify the Sequence Pattern
The given sequence is: ( \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, \ldots )
To recognize the pattern, let's rewrite the terms with common denominators for better clarity:
- ( \frac{1}{2} ) - ( \frac{1}{2} = \frac{1}{2} )
- ( 1 = \frac{2}{2} )
- ( \frac{3}{2} )
- ( 2 = \frac{4}{2} )
- ( \frac{5}{2} )
The numerators follow the sequence 1, 2, 3, 4, 5.
- Establish the Numerator Formula
The numerators of the sequence are consecutively increasing integers:
1, 2, 3, 4, 5, ...
This can be expressed as ( n ) where ( n ) is the index of the term:
- For ( n=1 ), numerator = 1
- For ( n=2 ), numerator = 2
- For ( n=3 ), numerator = 3
- For ( n=4 ), numerator = 4
- For ( n=5 ), numerator = 5
Thus, the formula for the numerator can be written as: $$ N(n) = n $$
- Identify the Denominator
The denominator in all terms is consistently 2, which can be expressed as: $$ D(n) = 2 $$
- Formulate the General Term
Combining the numerator and denominator, the general term ( a_n ) of the sequence can be expressed as: $$ a_n = \frac{N(n)}{D(n)} = \frac{n}{2} $$
The formula for the general term of the sequence is: $$ a_n = \frac{n}{2} $$
More Information
This formula shows that each term of the sequence is simply half of its index. For example, the first term (1) corresponds to ( a_1 = \frac{1}{2} ), the second term (2) to ( a_2 = 1 ), and so on. This approach can be generalized for any ( n ) in the sequence.
Tips
- A common mistake is to misidentify the pattern in the numerators. Ensure to examine each term thoroughly before drawing conclusions.
- Another mistake is forgetting to keep the denominator consistent. Always verify that it remains unchanged across terms.
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