Question
For positive integers n1,n2 the value of the expression (1+i)n1+(1+i3)n1+(1+i5)n2 + (1+i7)n2 where i=2√−1 is a real number if and only if
Answer: Option D
:
D
Using i3=−i,i5=iandi7=−i, we can write the given expression as
(1+i)n1+(1+i3)n1+(1+i5)n2+(1+i7)n2wherei=2√1=2[n1C0+n1C2(i)2+n1C4(i)4+n1C6(i)6+⋯⋯]+2[n2C0+n2C2(i)2+n2C4(i)4+n2C6(i)6+⋯⋯]=2[n1C0−n1C2+n1C4−n1C6+⋯⋯]+2[n2C0−n2C2+n2C4−n2C6+⋯⋯]
This is a real number irrespective of values of n1 and n2.
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:
D
Using i3=−i,i5=iandi7=−i, we can write the given expression as
(1+i)n1+(1+i3)n1+(1+i5)n2+(1+i7)n2wherei=2√1=2[n1C0+n1C2(i)2+n1C4(i)4+n1C6(i)6+⋯⋯]+2[n2C0+n2C2(i)2+n2C4(i)4+n2C6(i)6+⋯⋯]=2[n1C0−n1C2+n1C4−n1C6+⋯⋯]+2[n2C0−n2C2+n2C4−n2C6+⋯⋯]
This is a real number irrespective of values of n1 and n2.
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