The first five terms in a pattern are 0.25, 0.75, 1.25, 1.75, 2.25. If the pattern continues, which expressions can be used to find the nth term?
A. 0.25n
B. 0.5n – 0.25
C. -0.75n + 2
D. -0.5n + 0.75
The Correct Answer and Explanation is:
To find the nth term of the given pattern (0.25, 0.75, 1.25, 1.75, 2.25), we first need to analyze the sequence. Observing the terms, we see that they increase by (0.5) each time:
- (0.25) to (0.75) increases by (0.5)
- (0.75) to (1.25) increases by (0.5)
- (1.25) to (1.75) increases by (0.5)
- (1.75) to (2.25) increases by (0.5)
This consistent increase indicates a linear pattern. To express the nth term of the sequence, we can use the formula for the nth term of an arithmetic sequence, which is given by:
[
a_n = a_1 + (n – 1) \cdot d
]
where:
- (a_n) is the nth term,
- (a_1) is the first term,
- (d) is the common difference,
- (n) is the term number.
For this sequence:
- (a_1 = 0.25)
- (d = 0.5)
Substituting these values into the formula gives us:
[
a_n = 0.25 + (n – 1) \cdot 0.5
]
Simplifying this, we have:
[
a_n = 0.25 + 0.5n – 0.5 = 0.5n – 0.25
]
Thus, the expression that represents the nth term is (0.5n – 0.25), which corresponds to option B.
Now let’s evaluate the other options:
- A. (0.25n): This expression would yield (0.25), (0.50), (0.75), etc., which do not match the pattern.
- C. (-0.75n + 2): Plugging in values for (n) (e.g., (n = 1) gives (1.25), (n = 2) gives (0.5)), which do not match the terms in the pattern.
- D. (-0.5n + 0.75): This expression gives decreasing values, starting with (0.25) and heading downwards, which is incorrect for our sequence.
Therefore, the correct answer is B. (0.5n – 0.25), as it accurately describes the nth term of the sequence based on the identified pattern.