Answer:
[tex]a_n=-2a_{n-1}+1[/tex]
Step-by-step explanation:
Successive terms do not have a common difference or ratio, so the sequence is not arithmetic or geometric. We can get a clue as to its nature a couple of places:
- from the alternating signs, says there is a negative multiplier involved
- from the answer choices
It is helpful to look at the differences of terms:
-5-3 = -8
11 -(-5) = 16
-21-11 = -32
These have a common ratio of -2, so match an exponential form with a ratio of -2 (the last answer choice). Trying that choice with n=1, 2, 3, 4 gives ...
a[1] = 3
a[2] = -2(3) +1 = -5
a[3] = -2(-5) +1 = 11
a[4] = -2(11) +1 = -21
These match the given sequence, so the appropriate recursive formula is ...
[tex]\boxed{a_n=-2a_{n-1}+1}[/tex]
