Given
The sequence, 8, 5, 2, -1.
To find: Which of the following functions best defines this sequence?
a) f(1) = 8, f(n + 1) = f(n) + 5; for n=1,2,3,4,...
b) f(1) = 8, f(n + 1) = f(n) - 5; for n=1,2,3,4,...
c) f(1) = 8, f(n + 1) = f(n) - 3; for n=1,2,3,4,...
d) f(1) = 8, f(n + 1) = f(n) + 3; for n=1,2,3,4,...
Explanation:
It is given that,
The first four terms of a sequence is, 8, 5, 2, -1.
Since,
[tex]\begin{gathered} 5-8=-3 \\ 2-5=-3 \end{gathered}[/tex]Then, the above sequence is an arithmetic sequence.
That implies,
[tex]\begin{gathered} f(n)=f(1)+(n-1)d \\ =8+(n-1)(-3) \end{gathered}[/tex]Therefore, for n=1,2.
[tex]\begin{gathered} f(1)=8 \\ f(2)=8+(2-1)(-3) \\ =8-3 \\ =f(1)-3 \end{gathered}[/tex]Then,
[tex]f(n+1)=f(n)-3[/tex]Final result: Hence, the answer is option c).
