College

The pattern of numbers below is an arithmetic sequence:

\[ 14, 24, 34, 44, 54, \ldots \]

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 :

We start with the arithmetic sequence:

$$14,\ 24,\ 34,\ 44,\ 54,\ \ldots$$

Step 1. Identify the first term:
$$f(1)=14.$$

Step 2. Calculate the common difference by subtracting the first term from the second term:
$$24 - 14 = 10.$$

Step 3. Write the recursive function for an arithmetic sequence:
$$f(n+1) = f(n) + \text{common difference}.$$

Substitute the known common difference and the first term:
$$f(n+1) = f(n) + 10 \quad \text{with} \quad f(1)=14.$$

Thus, the correct recursive function is:

$$\textbf{f(n+1)=f(n)+10 where f(1)=14.}$$

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