Answer :
To solve this problem, we need to understand that the sequence provided is an arithmetic sequence. In an arithmetic sequence, each term is obtained by adding a constant number, known as the common difference, to the previous term.
Let's break down the given sequence:
The sequence is: 14, 24, 34, 44, 54, ...
To find the common difference, we subtract the first term from the second term:
Common difference = 24 - 14 = 10
Now that we know the common difference is 10, we need to find which recursive function correctly describes this sequence using the common difference and the first term.
Let's analyze the options:
1. The common difference is 1, so the function is [tex]\( f(n+1) = f(n) + 1 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
2. The common difference is 4, so the function is [tex]\( f(n+1) = f(n) + 4 \)[/tex] where [tex]\( f(1) = 10 \)[/tex].
3. The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
4. The common difference is 14, so the function is [tex]\( f(n+1) = f(n) + 14 \)[/tex] where [tex]\( f(1) = 10 \)[/tex].
Given that the common difference is 10 and the first term [tex]\( f(1) \)[/tex] of the sequence is 14, option 3 fits:
The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
This correctly describes the recursive function for the arithmetic sequence given.
Let's break down the given sequence:
The sequence is: 14, 24, 34, 44, 54, ...
To find the common difference, we subtract the first term from the second term:
Common difference = 24 - 14 = 10
Now that we know the common difference is 10, we need to find which recursive function correctly describes this sequence using the common difference and the first term.
Let's analyze the options:
1. The common difference is 1, so the function is [tex]\( f(n+1) = f(n) + 1 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
2. The common difference is 4, so the function is [tex]\( f(n+1) = f(n) + 4 \)[/tex] where [tex]\( f(1) = 10 \)[/tex].
3. The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
4. The common difference is 14, so the function is [tex]\( f(n+1) = f(n) + 14 \)[/tex] where [tex]\( f(1) = 10 \)[/tex].
Given that the common difference is 10 and the first term [tex]\( f(1) \)[/tex] of the sequence is 14, option 3 fits:
The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
This correctly describes the recursive function for the arithmetic sequence given.
