Question

3. For each sequence, determine whether it could be geometric, arithmetic, or

neither. Next, match each sequence with one of the recursive definitions.
Note that only the part of the definition showing the relationship between
the current term and the previous term is given so as not to give away the
solutions. (Lesson 1-5)
a. 3, 15, 75, 375
A. a(n) = a(n-1)
b. 18, 6, 2,
c. 1, 2, 4, 7
d. 17, 13, 9, 5
B. b(n)= b(n-1)-4
C. c(n) = 5 c(n-1)
D. d(n)=d(n-1) + n-1

Answer 1

Let's analyze each sequence and determine whether it could be geometric, arithmetic, or neither:

a. 3, 15, 75, 375

- This sequence is geometric because each term is obtained by multiplying the previous term by 5 (3 * 5 = 15, 15 * 5 = 75, and so on).

b. 18, 6, 2

- This sequence is arithmetic because each term is obtained by subtracting 12 from the previous term (18 - 12 = 6, 6 - 4 = 2).

c. 1, 2, 4, 7

- This sequence is neither geometric nor arithmetic because there is no consistent common ratio or common difference between the terms.

d. 17, 13, 9, 5

- This sequence is arithmetic because each term is obtained by subtracting 4 from the previous term (17 - 4 = 13, 13 - 4 = 9, and so on).

Now, let's match each sequence with the appropriate recursive definition:

a. 3, 15, 75, 375 (Geometric)

- Recursive Definition: C. c(n) = 5 c(n-1)

b. 18, 6, 2 (Arithmetic)

- Recursive Definition: B. b(n) = b(n-1) - 4

c. 1, 2, 4, 7 (Neither)

- No recursive definition provided because it's neither geometric nor arithmetic.

d. 17, 13, 9, 5 (Arithmetic)

- Recursive Definition: D. d(n) = d(n-1) + n-1

I hope this helps clarify the types of sequences and their respective recursive definitions!