Convert the recursive from a↓1 = 3, a↓n = -2a↓n-1 to the explicit form.
a. a↓n = 3(-2)^n-1
b. a↓n = -2(3)^n+1
c. a↓n = -2(3)^n-1
d. a↓n = 3(-2)^n
The recursive formula is defined as a↓1 = 3 and a↓n = -2a↓(n-1).
To find the explicit form, we can start by substituting the recursive formula for a↓n into the recursive formula for a↓(n-1).
Starting with a↓2 = -2a↓1, we substitute a↓1 = 3:
a↓2 = -2(3)
Continuing this process, we can find a↓3, a↓4, and so on:
a↓3 = -2(a↓2) = -2(-2(3))
a↓4 = -2(a↓3) = -2(-2(-2(3)))
a↓5 = -2(a↓4) = -2(-2(-2(-2(3))))
Observing this pattern, we can come up with the explicit form a↓n = (-2)^(n-1)(3).
So, the correct answer is d. a↓n = 3(-2)^n.