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Mathematics
Problem
For positive integers
n
1
,
n
2
n_1, n_2
n
1
,
n
2
, the value of expression
(
1
+
i
)
n
1
+
(
1
+
i
3
)
n
1
+
(
1
+
i
5
)
n
2
+
(
1
+
i
7
)
n
2
(1+i)^{n_1} + (1+i^3)^{n_1} + (1+i^5)^{n_2} + (1+i^7)^{n_2}
(
1
+
i
)
n
1
+
(
1
+
i
3
)
n
1
+
(
1
+
i
5
)
n
2
+
(
1
+
i
7
)
n
2
, where
i
=
−
1
i = \sqrt{-1}
i
=
−
1
is a real number, if and only if (B)
n
1
=
n
2
−
1
n_1 = n_2 - 1
n
1
=
n
2
−
1
(A)
n
1
=
n
2
+
1
n_1 = n_2 + 1
n
1
=
n
2
+
1
(D)
n
1
>
0
,
n
2
>
0
n_1 > 0, n_2 > 0
n
1
>
0
,
n
2
>
0
(C)
n
1
=
n
2
n_1 = n_2
n
1
=
n
2
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