Initial work

CS1029/practical-2023-10-20/sequences

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+1. a) 3
+   b) -1
+   c) 787
+   d) 2639
+
+2. a) 128
+   b) 7
+   c) 2
+   d) -256
+
+3. a) 1, -2, 4, -8
+   b) 3, 3, 3, 3
+   c) 8, 13, 23, 71
+   d) 2, 0, 8, 0
+
+4. a) 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
+   b) 0, 0, 0, 1, 1, 1, 2, 2, 2, 3
+   c) 1, 1, 3, 3, 5, 5, 7, 7, 9, 9
+   d) 3, 6, 12, 24, 48, 96, 192, 384, 768, 1336
+
+5. a_0 = 3, a_n = a_{n - 1} + 2 (i.e. the odd integers starting at 3)
+   the primes starting at three
+   - one more
+
+6. a) 2, 12, 72, 432, 2592
+   b) 2, 4, 16, 256, 65536
+   c) 1, 2, 5, 11, 26
+   d) 1, 1, 6, 27, 204
+
+7. a) a_n = -3a_{n-1} + 4a_{n-2} = -3(0) + 4(0) = 0 = a_n
+   b) a_n = -3(1) + 4(1) = 4 - 3 = 1 = a_n
+   c) a_n = -3((-4)^{n-1}) + 4((-4)^{n-2}) = -3(-4)(-4^{n-2}) + 4((-4)^{n-2}) = (-4)^{n-2}((-3)(-4) + 4) = (-4)^{n-1}(-3 - 1) = (-4)^n = a_n
+   d) a_n = -3(2*(-4)^{n-1} + 3) + 4(2 * (-4)^{n-2} + 3) = -6(-4)^{n-1} - 9 - 2(-4)^{n-1} + 12 = -8(-4)^{n-1} + 3 = 2(-4)^n + 3 = a_n
+
+8.