Adv Alegbra HW Solutions 5 - 5(ii Prove that Ramanujans...

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
( n + 1 )
2
− ( n + 1 ) =
1
2
n ( n + 1 ).
But | S | is the sum we are seeking.
1
2
3
4
5 1 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