Graphics Reference
In-Depth Information
Answers and comments
1
n
,2
n
, 1/2
,
to
n
1 places of decimals, (
1)
,
n
/(
n
1).
4
(a) (b) (c) (d)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
5
(a)
(b)
(i)
(ii)
(iii)
(iv)
6 (i) Yes, (
.
Monotonic decreasing: (i) yes; (ii) yes (1/
n
); (iii) yes, always,
a
1/
n
); (ii) yes; (iii) no, (
n
); (iv) yes, always,
a
; (iv) no,
(
n
).
7 (i) Yes; (ii) yes, because if there were only a finite number, the greatest
would bean uppr bound.
8 (i) A finite list, (ii) not in order, (iii) not in original sequence when
n
is
odd, (iv) not in original sequence when
n
1.
9 (i) 1, 4, 5, 8, 9, . . ., (ii) 2, 3, 6, 7, 10, . . ., (iii) 1, 3, 5, 7, 9, . . ., (iv) 2, 4, 6, 8,
10, . . ..
10
(i) (a) No. The
n
form an increasing sequence. (b) No. The
n
form a
strictly increasing sequence. (c) 5. (d)
a
, an integer greater than
101 may also beused.
(ii) (
n
) is strictly increasing, so
i
j
n
n
. But the sequence (
a
)
is arbitrary, so the subsequence need not be increasing.
11 (i) Yes, see qn 4(iv); (ii) not for a constant sequence.
12 Yes, the bounds for the sequence are certainly bounds for any
subsequence.
13
(i) If the subsequence is bounded above by
U
and bounded below
by
L
, then the sequence has max(
a
,
U
) as an upper bound and
) as a lower bound. 'max' is formally defined at qn 6.44.
(ii) As (i) with upper bound max(
a
min(
a
,
L
,
a
,
U
), lower bound
min(
a
,
a
,
L
).
