Answer :
To solve the problem, we first recognize that an arithmetic sequence increases by a constant difference (called the common difference). Here is a detailed step-by-step explanation:
1. The terms given are:
[tex]$$14, \; 24, \; 34, \; 44, \; 54, \ldots$$[/tex]
2. To determine the common difference, subtract the first term from the second term:
[tex]$$24 - 14 = 10.$$[/tex]
3. Check that this difference is the same between other consecutive terms:
[tex]$$34 - 24 = 10, \quad 44 - 34 = 10, \quad 54 - 44 = 10.$$[/tex]
Since all differences are equal, the common difference is indeed [tex]$10$[/tex].
4. In an arithmetic sequence, the recursive formula is given by:
[tex]$$f(n+1) = f(n) + d,$$[/tex]
where [tex]$d$[/tex] is the common difference. With [tex]$d = 10$[/tex] and the first term [tex]$f(1) = 14$[/tex], the recursive function becomes:
[tex]$$f(n+1) = f(n) + 10 \quad \text{with} \quad f(1) = 14.$$[/tex]
5. Comparing with the provided options, the statement that correctly describes this recursive function is:
[tex]$$\text{"The common difference is 10, so the function is } f(n+1)=f(n)+10 \text{ where } f(1)=14\text{."}$$[/tex]
Thus, the correct answer is the third option.
1. The terms given are:
[tex]$$14, \; 24, \; 34, \; 44, \; 54, \ldots$$[/tex]
2. To determine the common difference, subtract the first term from the second term:
[tex]$$24 - 14 = 10.$$[/tex]
3. Check that this difference is the same between other consecutive terms:
[tex]$$34 - 24 = 10, \quad 44 - 34 = 10, \quad 54 - 44 = 10.$$[/tex]
Since all differences are equal, the common difference is indeed [tex]$10$[/tex].
4. In an arithmetic sequence, the recursive formula is given by:
[tex]$$f(n+1) = f(n) + d,$$[/tex]
where [tex]$d$[/tex] is the common difference. With [tex]$d = 10$[/tex] and the first term [tex]$f(1) = 14$[/tex], the recursive function becomes:
[tex]$$f(n+1) = f(n) + 10 \quad \text{with} \quad f(1) = 14.$$[/tex]
5. Comparing with the provided options, the statement that correctly describes this recursive function is:
[tex]$$\text{"The common difference is 10, so the function is } f(n+1)=f(n)+10 \text{ where } f(1)=14\text{."}$$[/tex]
Thus, the correct answer is the third option.
