Question

A sequence is defined recursively below.

an=an-1-5;a1=-7
Which function can be used to determine the nth term of the sequence?
A. f(n)= -5n-7
B. f(n)= -2n-7
C. f(n)= -5n-2
D. f(n)= -2n-5

Answer 1

To find the nth term for the recursive sequence defined as a_n = a_{n-1} - 5, with the first term a_1 = -7, the correct function is f(n) = -5n - 2. This can be determined by finding a pattern in the differences between terms and applying the initial condition a_1 = -7 to develop an explicit formula.

Step-by-step explanation:

To find the function that can determine the nth term of the sequence, we first consider the recursive formula given: an = an-1 - 5 with the initial term a1 = -7.

The sequence is decreasing by 5 each time, which indicates that our function will involve the term -5n. We can find the explicit formula by observing the sequence:

• For n = 1, a1 = -7.
• For n = 2, a2 = a1 - 5 = -7 - 5 = -12.
• For n = 3, a3 = a2 - 5 = -12 - 5 = -17.

We can see that we are adding an additional -5 for each increase in n.

The explicit formula would be f(n) = -7 - 5(n - 1). Simplifying this, we get: f(n) = -7 - 5n + 5 = -5n - 2.

Therefore, the function to determine the nth term of the sequence is f(n) = -5n - 2, which is option B.