Problem sheet 5 29. Can you find a recurrence relation which

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Problem sheet 5
29. Can you find a recurrence relation which generates each of the following sequences (not
forgetting to mention the seed in each case)?
(a) 1, 1, 1, 1, . . .
(b) 2, 32 , 56 , 85 , . . .
(c) 3, 9, 81, 6561, . . .
(d) 4, 9, 19, 39, . . .
(e) 5, 3, 4, 32 , . . .
(f) 6, 4, 6, 4, . . .
(g) 7, − 76 , − 13
, − 19
,...
6
13
(h) 8, 4, 2, 1, . . .
(i) 9, 20, 31, 42, . . .
(j) 10, −40, 160, −640, . . .
(Note that these are not necessarily in order of difficulty, so if one exercise seems difficult,
skip to the next one and return later. Moreover, there may, in some cases, be more than one
way to express your answer.)
30. Which of the sequences in exercise 29 are increasing, which are decreasing and which are
neither. Justify your answer in each case.
31. Which of the sequences in exercise 29 are strictly increasing, which are strictly decreasing
and which are neither. Justify your answer in each case.
32. Which of the sequences in exercise 29 are monotone and which are not. Justify your answer
in each case.
33. Which of the sequences in exercise 29 are strictly monotone and which are not. Justify your
answer in each case.
34. Which of the sequences in exercise 29 are bounded above, which are bounded below, and
which are neither. Justify your answer in each case.
35. Which of the sequences in exercise 29 are bounded and which are not. Justify your answer
in each case.
36. Show that the formula given on page 11 for the Fibonacci sequence generates the correct
values for n = 1, 2, 3.
37. Show that if ai = kri were a solution for the Fibonacci sequence given
by ai =
a + ai−2 ,
√ i−1
√
1− 5
1+ 5
a1 = 1, a2 = 1, where k and r are constants, then r must be either 2 or 2 (which are
the solutions of the characteristic equation, x2 − x − 1 = 0).
√
√
38. Let p = 1+2 5 and q = 1−2 5 be the two solutions of the characteristic equation just mentioned
(where p > q). Show that p + q = 1 and pq = −1.
39. Expand (p2 − q 2 )(p + q) and hence show that p3 − q 3 = p2 − q 2 + p − q.
40. Show, likewise, that p4 − q 4 = p3 − q 3 + p2 − q 2
41. Show, in general, that pi − q i = pi−1 − q i−1 + pi−2 − q i−2 , where i ≥ 2.
n √ n √ n o
1+ 5
. Show that b1 = b2 = 1. (Have you seen this before?)
42. Let bn = √15
− 1−2 5
2
43. Finally prove that bn = an ∀n
44. The Lucas numbers are given by ai = ai−1 + ai−2 , a1 = 2, a2 = 1. Write down the first ten
terms. Try to find ai explicitly.
45. Construct a sequence of numbers from our set, S, which converges to 25 . Calculate the first
eight terms of this sequence.
46. Repeat exercise 45 for the fraction
25
.
17
47. Repeat exercise 45 for the irrational number
48. Repeat exercise 45 for the irrational number
√
√
3.
5.
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