Home
Class 12
MATHS
If n1, n2 are positive integers, then (1...

If `n_1, n_2` are positive integers, then `(1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i_5)^(n_2) + (1 + i^7)^(n_2)` is real if and only if :

Text Solution

Verified by Experts

`(1+i)^(n_(1))+(1+i^(3))^(n_(1)) + (1+i^(5))^(n_(2)) + (1+i^(7))^(n_(2))`
`= (1+i)^(n_(1)) +(1-i)^(n_(1)) +(1+i)^(n_(2))+(1-i)^(n_(2))`
`=[(1+i)^(n_(1))+bar(1+i)^(n_(1))]+[(1+i)^(n_(2))+bar(1+i)^(n_(2))]`
=Purely real
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE|Exercise Exercise 3.3|7 Videos

  • COMPLEX NUMBERS

    CENGAGE|Exercise Exercise 3.4|7 Videos

  • COMPLEX NUMBERS

    CENGAGE|Exercise Exercise 3.1|4 Videos

  • CIRCLES

    CENGAGE|Exercise Question Bank|32 Videos

  • CONIC SECTIONS

    CENGAGE|Exercise Solved Examples And Exercises|91 Videos

Similar Questions

Explore conceptually related problems

If n is a positive integer,then (1+i)^(n)+(1-1)^(n) is equal to

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 (a) n_1=n_2+1 (b) n_1=n_2-1 (c) n_1=n_2 (d) n_1 > 0, n_2 > 0

The smallest positive integer n forwhich ((1 + i)/(1 - i))^(n) = 1 is:

For a positive integer n, find the value of (1-i)^(n)(1-(1)/(i))^(n)

If n is any positive integer,write the value of (i^(4n+1)-i^(4n-1))/(2)

The expression (1+i)^(n_1)+(1+i^(3))^(n_(2)) is real iff