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Which recursive formula can be used to generate the sequence shown, where [tex]f(1) = 5[/tex] and [tex]n \geq 1[/tex]?

[tex]5, -1, -7, -13, -19, \ldots[/tex]

A. [tex]f(n+1) = f(n) + 6[/tex]

B. [tex]f(n) = f(n+1) - 6[/tex]

C. [tex]f(n+1) = f(n) - 6[/tex]

D. [tex]f(n+1) = -6f(n)[/tex]



Answer :

GinnyAnswer
To determine the appropriate recursive formula for the sequence [tex]\(5, -1, -7, -13, -19, \ldots \)[/tex], we need to observe how the terms change from one to the next.

First, let's note the initial values:
- The first term [tex]\( f(1) \)[/tex] is 5.

Next, let's find the difference between each consecutive term:
- The difference between the second term and the first term: [tex]\(-1 - 5 = -6\)[/tex].
- The difference between the third term and the second term: [tex]\(-7 - (-1) = -7 + 1 = -6\)[/tex].
- The difference between the fourth term and the third term: [tex]\(-13 - (-7) = -13 + 7 = -6\)[/tex].
- The difference between the fifth term and the fourth term: [tex]\(-19 - (-13) = -19 + 13 = -6\)[/tex].

We see that the sequence consistently decreases by 6 at each step. This means that to get the next term, we subtract 6 from the current term.

Therefore, the recursive formula that fits this pattern is:
[tex]\[ f(n+1) = f(n) - 6 \][/tex]

This means the correct formula to generate the given sequence is:
[tex]\[ \boxed{f(n+1) = f(n) - 6} \][/tex]

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