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5
(ii)
Prove that Ramanujan’s statement is false.
Solution.
One must pay attention to hypotheses. Consider
a
3
+
b
3
if
b
is negative:
728
=
12
3
+
(
−
10
3
)
=
9
3
+
(
−
1
)
3
.
1.11
Derive the formula for
∑
n
i
=
1
i
by computing the area
(
n
+
1
)
2
of a square
with sides of length
n
+
1 using Figure 1.1.
Solution.
Compute the area
A
of the square in two ways. On the one hand,
A
=
(
n
+
1
)
2
. On the other hand,
A
=
D
+
2

S

, where
D
is the diagonal
and
S
is the “staircase.” Therefore,

S
=
1
2
h
(
n
+
1
)
2
−
(
n
+
1
)
i
=
1
2
n
(
n
+
1
).
But

S

is the sum we are seeking.
1
2
3
4
51
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Figure 1.1
1
+
2
+···+
n
=
1
2
(
n
2
+
n
)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Figure 1.2
1
+
2
n
=
1
2
n
(
n
+
1
)
1.12
(i)
Derive the formula for
∑
n
i
=
1
i
by computing the area
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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