Question

2. (10 points) Find the first six terms of the sequence defined by each of these recurrence relations and intital conditions. (a) a

n
​
=−2a
n−1
​
,a
0
​
=−1 (b) a
n
​
=a
n−1
​
−a
n−2
​
,a
0
​
=2,a
1
​
=−1 (c) a
n
​
=3a
n−1
2
​
,a
0
​
=1 (d) a
n
​
=na
n−1
​
+a
n−2
2
​
,a
0
​
=−1,a
1
​
=0 (e) a
n
​
=a
n−1
​
−a
n−2
​
+a
n−3
​
,a
0
​
=1,a
1
​
=1,a
2
​
=2

Answer 1

The first six terms of the given recurrence relations are: (a) -1, 2, -4, 8, -16, 32. (b) 2, -1, -3, -2, 1, 3. (c) 1, 3, 27, 2187, 43046721, 1853020188851841. (d) -1, 0, -2, -4, -1, -4. (e) 1, 1, 2, 3, 6, 9.

Step-by-step explanation:

1. (a) The first six terms of the sequence defined by the recurrence relation an = -2an-1, a0 = -1 are: -1, 2, -4, 8, -16, 32.
2. (b) The first six terms of the sequence defined by the recurrence relation an = an-1 - an-2, a0 = 2, a1 = -1 are: 2, -1, -3, -2, 1, 3.
3. (c) The first six terms of the sequence defined by the recurrence relation an = 3an-12, a0 = 1 are: 1, 3, 27, 2187, 43046721, 1853020188851841.
4. (d) The first six terms of the sequence defined by the recurrence relation an = nan-1 + an-22, a0 = -1, a1 = 0 are: -1, 0, -2, -4, -1, -4.
5. (e) The first six terms of the sequence defined by the recurrence relation an = an-1 - an-2 + an-3, a0 = 1, a1 = 1, a2 = 2 are: 1, 1, 2, 3, 6, 9.