Section 5.2
135
30.
al
=
1,
a2
=
al
=
1,
a3
=
al
+
a2
=
2,
a4
=
al
+
a2
+
a3
=
4,
a5
=
al
+
a2
+
a3
+
a4
=
8,
a6
=
al
+
a2
+
a3
+
a4
+
a5
=
16. We guess
an
=
2
n

2
for
n
:::::
2 and prove this result by
mathematical induction (strong form).
For
n
=
2,
a2
=
1 while 2
2

2
=
2
0
=
1 also, so the formula is correct. Now assume that
k
> 2 and
the formula is true for all
n,
2 S
n
<
k.
Then
kl
kl
ak
=
L
ai
=
al
+
L
ai
i=1
i=2
kl
1
+
L
2
i

2
using
al
=
1 and the induction hypothesis
i=2
1(1 
2
k

2
)
=
1
+
1
+
2
+
4
+ .
.. +
2
k

3
=
1
+
=
1
+
(2
k

2

1)
=
2
k

2
12
which is the desired formula with
n
=
k.
(Note that we used formula (8) to sum
k

2 terms of the
geometric sequence with
a
=
1,
r
=
2 at the second last step.) We conclude that the formula is true
for all
n
::::: 2, by the Principle of Mathematical Induction.
7
(18
18
)
7 (2
54
_1)
31. (a) The sum is 1024
1 _ 8
=
2
10
7
=
244 
T
10
~
244.
(b) [BB]
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 Summer '10
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 Graph Theory, Mathematical Induction, Natural number, 2 K, 1 K, Geometric progression, 2k, 2 k

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