The pattern of numbers below is an arithmetic sequence:
[tex]$14,24,34,44,54, \ldots$[/tex]
Which statement describes the recursive function used to generate the sequence?
A. The common difference is 1, so the function is [tex]$f(n+1)=f(n)+1$[/tex] where [tex]$f(1)=14$[/tex].
B. The common difference is 4, so the function is [tex]$f(n+1)=f(n)+4$[/tex] where [tex]$f(1)=10$[/tex].
C. The common difference is 10, so the function is [tex]$f(n+1)=f(n)+10$[/tex] where [tex]$f(1)=14$[/tex].
D. The common difference is 14, so the function is [tex]$f(n+1)=f(n)+14$[/tex] where [tex]$f(1)=10$[/tex].
Answer (2)
Determine the common difference by subtracting consecutive terms: d = 24 − 14 = 10 .
Write the recursive function as f ( n + 1 ) = f ( n ) + d , which becomes f ( n + 1 ) = f ( n ) + 10 .
State the initial condition: f ( 1 ) = 14 .
Combine the recursive function and the initial condition: f ( n + 1 ) = f ( n ) + 10 where f ( 1 ) = 14 , so the answer is f ( n + 1 ) = f ( n ) + 10 where f ( 1 ) = 14 .
Explanation
Analyzing the Sequence Let's analyze the given arithmetic sequence: 14 , 24 , 34 , 44 , 54 , … . We need to find the recursive function that generates this sequence. A recursive function defines each term in the sequence based on the previous term.
General Recursive Function The general form of a recursive function for an arithmetic sequence is f ( n + 1 ) = f ( n ) + d , where d is the common difference between consecutive terms.
Finding the Common Difference To find the common difference d , we subtract a term from its subsequent term. For example, 24 − 14 = 10 . So, the common difference d is 10.
Writing the Recursive Function Now we can write the recursive function as f ( n + 1 ) = f ( n ) + 10 . This means each term is obtained by adding 10 to the previous term.
Identifying the Initial Condition The first term of the sequence is f ( 1 ) = 14 . This is our initial condition, which tells us where the sequence starts.
Complete Recursive Definition Combining the recursive function and the initial condition, we get the complete recursive definition for the sequence: f ( n + 1 ) = f ( n ) + 10 where f ( 1 ) = 14 . This matches the third option provided.
Final Answer Therefore, the correct statement is: The common difference is 10, so the function is f ( n + 1 ) = f ( n ) + 10 where f ( 1 ) = 14 .
Examples
Arithmetic sequences and recursive functions are used in various real-life scenarios, such as calculating simple interest, predicting population growth, and 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 initially and earn $10 every year, the balance each year forms an arithmetic sequence: $100, $110, 120 , an d soo n . T h erec u rs i v e f u n c t i o nh e lp s t oe a s i l yc a l c u l a t e t h e ba l an ce f or an y g i v e n ye a r : f(n+1) = f(n) + 10 w h ere f(1) = 100$.
The recursive function for the arithmetic sequence 14, 24, 34, 44, 54,... is f ( n + 1 ) = f ( n ) + 10 with the initial condition f ( 1 ) = 14 . Thus, the correct choice is option C.
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