Write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. (Assume that n begins with 1.)
a (subtext) 1 =4,a (subtext) (k+1) =−a (subtext) k
a. a (subtext) n =4(−1)^n−1
b. a (subtext) n =4n
c. a (subtext) n =(−4)^n−1
d. a (subtext) n =(−4)^n
e. a (subtext) n =4(−1)^n
since you keep multiplying by -1, there will be (n-1) of them by the nth term.
To find the first five terms of the sequence defined recursively, we can follow the recursive rule.
Given the recursive definition:
a₁ = 4
aₖ₊₁ = -aₖ
We can calculate the first few terms manually:
a₁ = 4
a₂ = -a₁ = -4
a₃ = -a₂ = 4
a₄ = -a₃ = -4
a₅ = -a₄ = 4
Therefore, the first five terms of the sequence are: 4, -4, 4, -4, 4.
Now, let's find the nth term of the sequence as a function of n.
We can observe that the sequence alternates between 4 and -4. Notice that the sign of each term follows a pattern: positive for even values of n and negative for odd values of n.
Therefore, the nth term of the sequence can be represented as:
aₙ = 4 * (-1)^(n-1)
Hence, the correct answer is:
aₙ = 4 * (-1)^(n-1), which corresponds to option a.