Complex NumbersHard
Question
For positive integers n1, n2 the value of expression (1+ i)n1 + (1 + i3)n1 + (1 + i5)n2 + (1 + i7)n2, here i = √- 1 is a real number, if and only if
Options
A.n1 = n2 + 1
B.n1 = n2 - 1
C.n1 = n2
D.n1 > 0, n2 > 0
Solution
(1 + i)n1 + (- i)n1 + (1 + i)n1 (1 - i)n1 +
= [n1C0 + n1C1i1 + n1C2i2 + n1C3i3 + .......]
+ [n1C0 - n1C1i - n1C2i + n1C3i3+ .......]
+ [n2C0 + n2C1i + n2C2i2 + n2C3i3+ .......]
+ [n2C0 - n2C1i + n2C2i2 - n2C3i3+ .......]
+ [n1C0 + n1C2i2 + n1C4i4+ .......]
+ 2[n2C0 + n2C2i2 + n2C4i4+ .......]
= 2[n1C0 - n1C2 + n1C4 - ......]
+ 2[n2C0 - n2C2 + n2C4 - ......]
This is a real number irrespective of the values of n1 and n2.
Alternate Solution
{(1 + i)n1 + (1 - i)n1} + {(1 + i)n2 + (1 - i)n2}
⇒ a real number for all n1 and n2 ∈ R.
[∵ z +
= 2Re(z) ⇒ (1 + i)n1 + (1 - i)n1 is re4al number for all n ∈ R]
Hence, option (d) is the best option.
= [n1C0 + n1C1i1 + n1C2i2 + n1C3i3 + .......]
+ [n1C0 - n1C1i - n1C2i + n1C3i3+ .......]
+ [n2C0 + n2C1i + n2C2i2 + n2C3i3+ .......]
+ [n2C0 - n2C1i + n2C2i2 - n2C3i3+ .......]
+ [n1C0 + n1C2i2 + n1C4i4+ .......]
+ 2[n2C0 + n2C2i2 + n2C4i4+ .......]
= 2[n1C0 - n1C2 + n1C4 - ......]
+ 2[n2C0 - n2C2 + n2C4 - ......]
This is a real number irrespective of the values of n1 and n2.
Alternate Solution
{(1 + i)n1 + (1 - i)n1} + {(1 + i)n2 + (1 - i)n2}
⇒ a real number for all n1 and n2 ∈ R.
[∵ z +
= 2Re(z) ⇒ (1 + i)n1 + (1 - i)n1 is re4al number for all n ∈ R]Hence, option (d) is the best option.
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