For positive integer `n_1,n_2` the value of the expression `(1+i)^(n1) +(1+i^3)^(n1) (1+i^5)^(n2) (1+i^7)^(n_20),` where `i=sqrt-1,` is a real number if and only if

Options

A. `n_1=n_2+1`
B. `n_1=n_2-1`
C. `n_1=n_2`
D. `n_1 gt 0, n_2 gt 0 `

Answer

D

Explanation

`(1+i)^(n_1)+(1-i)^(n_1)+(1+i)^(n_2)+(1-i)^(n_2)`
`=""^(n_1)C_0+""^(n_1)C_1i+""^(n_2)C_2i^2+""^(n_2)C_3i^3+.....]`
`[+""^(n_1)C_0-""^(n_1)C_1i+""^(n_2)C_2i^2-""^(n_2)C_3i^3+.....]`
`[+""^(n_1)C_0+""^(n_1)C_1i+""^(n_2)C_2i^2+""^(n_2)C_3i^3+.....]`
`+[""^(n_1)C_0-""^(n_1)C_1i+""^(n_2)C_2i^2-""^(n_2)C_3i^3+.....]`
`=2[""^(n_1)C_0-""^(n_1)C_2i^2+""^(n_4)C_4i^4+.....]`
`+2[""^(n_1)C_0-""^(n_2)C_2i^2+""^(n_4)C_4i^4+.....]`
`=2[""^(n_1)C_0-""^(n_1)C_2+""^(n_1)C_4-....]+2[""^(n_2)C_0-""^(n_2)C_2+""^(n_2)C_2-....]`
This is a real number irrespective of the values of `n_1` and `n_2`
Alternate Solution
`{(1+i)^(n_1)+(1-i)^(n_1)}+[(1+i)^(n_2)+(1-i)^(n_2)}`
`rArr ` A real number for all `n_1` and `n_2 in R`
`[ therefore z+barz=2 Re (z) rArr (1+i)^(n_1)+(1-i)^(n-1)` is real number for all ` n in R]`