Determine the explicit form of the function represented by the sequence below.
$3,7,11,15$
A. $f(n)=3 n-4$
B. $f(n)=n+4$
C. $f(n)=4 n+3$
D. $f(n)=4 n-1$
Answer (2)
The sequence is arithmetic with a common difference of 4.
The general form of an arithmetic sequence is f ( n ) = a + ( n − 1 ) d .
Substituting a = 3 and d = 4 , we get f ( n ) = 3 + ( n − 1 ) 4 = 4 n − 1 .
The explicit form of the function is f ( n ) = 4 n − 1 .
Explanation
Understanding the Problem We are given the sequence 3 , 7 , 11 , 15 and asked to find the explicit form of the function that represents this sequence. The explicit form will allow us to find any term in the sequence directly by substituting the term number n into the formula.
Finding the Common Difference First, let's check if the sequence is arithmetic by finding the difference between consecutive terms:
7 − 3 = 4 11 − 7 = 4 15 − 11 = 4
Since the difference between consecutive terms is constant and equal to 4, the sequence is arithmetic with a common difference of 4.
Determining the Explicit Form The general form of an arithmetic sequence is f ( n ) = a + ( n − 1 ) d , where a is the first term and d is the common difference. In our case, a = 3 and d = 4 . Substituting these values into the general form, we get:
f ( n ) = 3 + ( n − 1 ) 4 f ( n ) = 3 + 4 n − 4 f ( n ) = 4 n − 1
Verification Now, let's check if our explicit form is correct by plugging in n = 1 , 2 , 3 , 4 :
For n = 1 , f ( 1 ) = 4 ( 1 ) − 1 = 3 For n = 2 , f ( 2 ) = 4 ( 2 ) − 1 = 7 For n = 3 , f ( 3 ) = 4 ( 3 ) − 1 = 11 For n = 4 , f ( 4 ) = 4 ( 4 ) − 1 = 15
The explicit form correctly generates the given sequence.
Final Answer Comparing our explicit form f ( n ) = 4 n − 1 with the given options, we see that it matches option D.
Therefore, the correct answer is D. f ( n ) = 4 n − 1 .
Examples
Arithmetic sequences are useful in many real-life situations, such as calculating simple interest, predicting the cost of items that increase at a constant rate, or determining the number of seats in an auditorium where each row has a fixed number of additional seats compared to the previous row. For example, if you deposit $100 into a savings account that earns $5 simple interest each year, the sequence of your account balance each year (100, 105, 110, ...) forms an arithmetic sequence.
The sequence 3 , 7 , 11 , 15 is an arithmetic sequence with a common difference of 4, leading to the explicit function f ( n ) = 4 n − 1 . The calculation confirms that this function accurately produces the terms of the sequence. Thus, the correct choice is D.
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