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National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1964 AMC 12/AHSME
34
34
Part of
1964 AMC 12/AHSME
Problems
(1)
Imaginary i
Source: 1964 AHSME Problem 34
8/26/2013
If
n
n
n
is a multiple of
4
4
4
, the sum
s
=
1
+
2
i
+
3
i
2
+
.
.
.
+
(
n
+
1
)
i
n
s=1+2i+3i^2+ ... +(n+1)i^{n}
s
=
1
+
2
i
+
3
i
2
+
...
+
(
n
+
1
)
i
n
, where
i
=
−
1
i=\sqrt{-1}
i
=
−
1
, equals:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
+
i
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
2
(
n
+
2
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
2
(
n
+
2
−
n
i
)
<span class='latex-bold'>(A)</span>\ 1+i\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2}(n+2) \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}(n+2-ni) \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
+
i
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
1
(
n
+
2
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
1
(
n
+
2
−
ni
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1
2
[
(
n
+
1
)
(
1
−
i
)
+
2
]
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
1
8
(
n
2
+
8
−
4
n
i
)
<span class='latex-bold'>(D)</span>\ \frac{1}{2}[(n+1)(1-i)+2]\qquad<span class='latex-bold'>(E)</span>\ \frac{1}{8}(n^2+8-4ni)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
1
[(
n
+
1
)
(
1
−
i
)
+
2
]
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
8
1
(
n
2
+
8
−
4
ni
)
AMC