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Question

For positive integers the value of the expression (1+i)n1+(1+i3)n1+(1+i5)n2+(1+i7)n2where i=1is a real number if and only if

A
n1=n2+1
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B
n1=n2+1
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C
n1=n2
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D
n1>0,n2>0
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Solution

The correct option is D n1>0,n2>0
Using i3=i,i5=i and i7=1, we can write the given expression as (1+i)n1+(1i)n1+(1+i)n2+(1i)n2
=2[n1c0+n1c4i4+.....]+2[n2c0+n2C2i2+n2C4i4.....]=2[n2C0n1C2+n1C4+....]+2[n2C0n2C2+n2C4+....]
This is a real number irrespective of the values of n1and n2.

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