if (1+i)(2+i)(3+i)(n+i)=a+ib then prove thAt 2*5*10*(n^2+i)=(a^2+b^2)
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Answer:
Step-by-step explanation:
Using:
|
z
|
1
⋅
|
z
|
2
⋅
...
⋅
|
z
|
n
=
|
z
1
⋅
z
2
⋅
...
⋅
z
n
|
⇒
|
z
|
2
1
⋅
|
z
|
2
2
⋅
...
⋅
|
z
|
2
n
=
|
z
1
⋅
z
2
⋅
...
⋅
z
n
|
2
And
|
z
|
2
=
z
⋅
¯
z
For the LHS:
|
(
1
+
i
)
(
2
+
i
)
(
3
+
i
)
...
.
(
n
+
i
)
|
2
=
|
1
+
i
|
2
|
2
+
i
|
2
|
3
+
i
|
2
...
.
|
n
+
i
|
2
=
(
1
+
i
)
¯¯¯¯¯¯¯¯¯¯¯
(
1
+
i
)
(
2
+
i
)
¯¯¯¯¯¯¯¯¯¯¯
(
2
+
i
)
(
3
+
i
)
¯¯¯¯¯¯¯¯¯¯¯
(
3
+
i
)
...
(
n
+
i
)
¯¯¯¯¯¯¯¯¯¯¯¯
(
n
+
i
)
=
(
1
+
1
)
(
4
+
1
)
(
9
+
1
)
...
(
n
2
+
1
)
=
2.5
.10
...
.
(
n
2
+
1
)
For the RHS:
|
a
+
i
b
|
2
=
a
2
+
b
2
∴
LHS = RHS
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