Question

The first five terms of a sequence are 7, 10, 13, 16, and 19. Which of the following functions define this sequence for all integers n ≥ 1?

A. f(n) = 4 + 3n
B. f(n) = 3n + 7
C. f(n) = 7(3)^(n-1)
D. f(n) = 4(3)^(n-1)

Answer 1

The correct answer is A.

Answer 2

The correct function defining the sequence 7, 10, 13, 16, and 19 for all integers n ≥ 1 is A. f(n) = 4 + 3n. This sequence represents an arithmetic progression with a first term of 7 and a common difference of 3.

Step-by-step explanation:

The student is asking for the correct function that defines the given sequence of numbers: 7, 10, 13, 16, and 19 for all integers n ≥ 1. By examining the pattern, we can see that the sequence is arithmetic because each term increases by a constant difference of 3. A general form for an arithmetic sequence is f(n) = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference. In this case, the first term a is 7, and the common difference d is 3, thus f(n) = 7 + (n - 1)×3.

By simplifying this, we have f(n) = 7 + 3n - 3 = 3n + 4. Now we need to compare this with the given options:

1. A. f(n) = 4 + 3n (Correct, as it matches the simplified form)
2. B. f(n) = 3n + 7 (Incorrect, this would start at 10 for n = 1)
3. C. f(n) = 7(3)^(n-1) (Incorrect, this represents a geometric sequence where each term is 3 times the previous one)
4. D. f(n) = 4(3)^(n-1) (Incorrect, same as above)

Therefore, the correct function is A. f(n) = 4 + 3n.