Example 2: The sequence a 1 , a 2 , a 3 ,... , an,... is such that
an =
an−1 − an−2
2 for all n > 2. If a 3 = 12 and a 4 = 5, what is a 2?
SOLUTION: Substitute the value of a 4 for an, the value of a 3 for an−1 and a 2
for an−2:
5 =
12 − a 2
2
10 = 12 – a 2
a 2 = 2
Example 3: In the sequence S, s 1 = 4, s 2 = 11, and s 3 = 18. Which of the
following could be the definition of the sequence?
A sn = sn−1 + 4
B sn = 2sn− 1 + 3
C sn = sn− 1 + 7
D sn = 2sn− 1 – 4
E sn = sn− 1 – 7
SOLUTION: Substitute the three values into each of the choices, and determine
which choice maintains the values for all three terms:
A: 11 = 4 + 4? No → Eliminate Choice A.
B: 11 = 2 4 + 3? Yes
18 = 2 11 + 3? No
→ Eliminate Choice B.
C: 11 = 4 + 7? Yes
18 = 11 + 7? Yes
→ Keep Choice C.
D: 11 = 2 4 – 4? No → Eliminate Choice D.
E: 11 = 4 – 7 No → Eliminate Choice E.
CHAPTER 11 ■ ALGEBRA 295
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