For positive integers n 1 ,n 2 the value of the expression ( 1 + i ) n 1 + ( 1 + i 3 ) n 1 + ( 1 + i 5 ) n 2 + ( 1 + i 7 ) n 2 where i= 2 √ − 1 is a real number if and only if Options: A . &nbsp n 1 = n 2 +1 B . &nbsp n 1 = n 2 -1 C . &nbsp n 1 = n 2 D . &nbsp n 1 >0, n 2 >0

Answer: Option D : D Using i 3 = − i , i 5 = i a n d i 7 = − i , we can write the given expression as ( 1 + i ) n 1 + ( 1 + i 3 ) n 1 + ( 1 + i 5 ) n 2 + ( 1 + i 7 ) n 2 w h e r e i = 2 √ 1 = 2 [ n 1 C 0 + n 1 C 2 ( i ) 2 + n 1 C 4 ( i ) 4 + n 1 C 6 ( i ) 6 + ⋯ ⋯ ] + 2 [ n 2 C 0 + n 2 C 2 ( i ) 2 + n 2 C 4 ( i ) 4 + n 2 C 6 ( i ) 6 + ⋯ ⋯ ] = 2 [ n 1 C 0 − n 1 C 2 + n 1 C 4 − n 1 C 6 + ⋯ ⋯ ] + 2 [ n 2 C 0 − n 2 C 2 + n 2 C 4 − n 2 C 6 + ⋯ ⋯ ] This is a real number irrespective of values of n 1 and n 2 .

For positive integers n1,n2 the value of the expression (1+i)n1+(1+

Discussion Forum : Complex Numbers

Question -

For positive integers n₁,n₂ the value of the expression $(1 + i)^{n_{1}}$ + $(1 + i^{3})^{n_{1}}$ + $(1 + i^{5})^{n_{2}}$ + $(1 + i^{7})^{n_{2}}$ where i= $\sqrt[2]{- 1}$ is a real number if and only if

Options:

A .

n_{1}

n_{2}

B .

n_{1}

n_{2}

-1

C .

n_{1}

n_{2}

D .

n_{1}

>0,

n_{2}

Answer: Option D
:
D

Using $i^{3} = - i, i^{5} = i a n d i^{7} = - i$ , we can write the given expression as
$(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}} w h e r e i = \sqrt[2]{1} = 2 [^{n_{1}} C_{0} +^{n_{1}} C_{2} (i)^{2} +^{n_{1}} C_{4} (i)^{4} +^{n_{1}} C_{6} (i)^{6} + \dots \dots] + 2 [^{n_{2}} C_{0} +^{n_{2}} C_{2} (i)^{2} +^{n_{2}} C_{4} (i)^{4} +^{n_{2}} C_{6} (i)^{6} + \dots \dots] = 2 [^{n_{1}} C_{0} -^{n_{1}} C_{2} +^{n_{1}} C_{4} -^{n_{1}} C_{6} + \dots \dots] + 2 [^{n_{2}} C_{0} -^{n_{2}} C_{2} +^{n_{2}} C_{4} -^{n_{2}} C_{6} + \dots \dots]$
This is a real number irrespective of values of $n_{1}$ and $n_{2}$ .

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