College

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 :

Certainly! Let's solve the problem step by step to identify the correct recursive function statement for the given arithmetic sequence: 14, 24, 34, 44, 54, ...

1. Identify the Sequence Type:
This is an arithmetic sequence, which means each term is obtained by adding a fixed number, called the common difference, to the previous term.

2. Find the Common Difference:
To find the common difference, subtract the first term from the second term in the sequence:
[tex]\( 24 - 14 = 10 \)[/tex]

Therefore, the common difference is 10.

3. Understand the Recursive Function:
A recursive function for an arithmetic sequence with the common difference [tex]\(d\)[/tex] is generally expressed as:
[tex]\( f(n+1) = f(n) + d \)[/tex]
In this case, given that the common difference [tex]\(d = 10\)[/tex], the function is:
[tex]\( f(n+1) = f(n) + 10 \)[/tex]

4. Identify the First Term:
The first term of the sequence is given as 14, so:
[tex]\( f(1) = 14 \)[/tex]

5. Choose the Correct Statement:
Now, we can choose the statement that matches our findings:
- The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].

Thus, the correct statement describing the recursive function used to generate the arithmetic sequence is:

"The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex]."