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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=21 is a real number if and only if


A

=+1

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B

=-1

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C

=

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D

>0,>0

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Solution

The correct option is D

>0,>0


Using i3=i,i5=i and i7=i, we can write the given expression as
(1+i)n1+(1+i3)n1+(1+i5)n2+(1+i7)n2 where i=21=2[n1C0+n1C2(i)2+n1C4(i)4+n1C6(i)6+]+2[n2C0+n2C2(i)2+n2C4(i)4+n2C6(i)6+]=2[n1C0n1C2+n1C4n1C6+]+2[n2C0n2C2+n2C4n2C6+]
This is a real number irrespective of values of n1 and n2.


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