IIT JEE , AIEEE Forum - Mathematics , Algebra

vinu_pee

for +ve n what is the value of (-1+i

3)³⁹ⁿ+(-1-

-3)⁹³ⁿ

vinu

Hi,namesake.

mul. & div. ur expression by 2^39n; (^ means power raised to)

u get , 2^39n[cis 39n {2# / 3}+cis 39n{4# / 3}]

( using prop's, (cis ?)^n=cisn? ; cis ?+cis @=cis( ?+@ ) ;)

u' ll ' ve ; 2^39n[cis 39n{2#}]

=2^39n[(cis 2#)^39n]

=2^39n[1]

So, the ans is ....... 2^39n.

titun

I think the answer provided by Vinu is wrong, it should be 2^39n + 2^93n (Here n must be an integer, otherwise there can't be any specific answer)

e.g. For n=1/3, the answer is 2^39n x w + 2^93n x w^2

For n being an improper fraction, the answer will be even more complicated.

So, there should be proper constraints on n (not just n>0) but n being a postive integer.

vinu

Oops,I'm so sorry, vinu_pee.

I didn't observe the power of 2nd term as 93n.

The ans is- 2^39n+2^93n.( as rightly mentioned by titun.)

edison

To simplify (-1+i

3)³⁹ⁿ+(-1-

-3)⁹³ⁿ

Following approach may be suggested.

Consider first term = (-1+i

3)³⁹ⁿ

Here -1+i

3 = (r cos

+i. r sin

), where r =

(1+3) =2

and tan

= -

3, find

to satisfy this equation

then use demoivre's theorem that is

(cos

+ i sin

)ⁿ = (cos n

+ i sin n

) = eⁱⁿ

similarly simplify second term and solve the expression.