Answer :
We are given the sequence:
$$14,\ 24,\ 34,\ 44,\ 54,\ \ldots$$
and we need to determine the recursive function that generates this sequence.
**Step 1: Identify the first term**
The first term is:
$$f(1) = 14.$$
**Step 2: Determine the common difference**
The common difference is found by subtracting the first term from the second term. That is,
$$\text{Common difference} = 24 - 14 = 10.$$
**Step 3: Write the recursive function**
Since an arithmetic sequence with a common difference of $d$ is defined recursively by
$$f(n+1) = f(n) + d,$$
substituting the common difference $d = 10$, we have:
$$f(n+1) = f(n) + 10.$$
**Step 4: Verify the function**
Starting from $f(1) = 14$, the next term would be:
$$f(2) = f(1) + 10 = 14 + 10 = 24,$$
confirming that the function correctly reproduces the sequence.
Thus, the recursive function is described by:
$$\text{The common difference is 10, so the function is } f(n+1)=f(n)+10 \text{ where } f(1)=14.$$
$$14,\ 24,\ 34,\ 44,\ 54,\ \ldots$$
and we need to determine the recursive function that generates this sequence.
**Step 1: Identify the first term**
The first term is:
$$f(1) = 14.$$
**Step 2: Determine the common difference**
The common difference is found by subtracting the first term from the second term. That is,
$$\text{Common difference} = 24 - 14 = 10.$$
**Step 3: Write the recursive function**
Since an arithmetic sequence with a common difference of $d$ is defined recursively by
$$f(n+1) = f(n) + d,$$
substituting the common difference $d = 10$, we have:
$$f(n+1) = f(n) + 10.$$
**Step 4: Verify the function**
Starting from $f(1) = 14$, the next term would be:
$$f(2) = f(1) + 10 = 14 + 10 = 24,$$
confirming that the function correctly reproduces the sequence.
Thus, the recursive function is described by:
$$\text{The common difference is 10, so the function is } f(n+1)=f(n)+10 \text{ where } f(1)=14.$$
