Posted in Math Club

Sum of different powers of imaginary numbers with different coefficients

Practice for Math Club

Many member noticed that there is a pattern within the power of i.

I = sqrt(-1)

I² = -1

I³ = -sqrt(-1) = –I

I^4 = +1

 

And they separated i, i³ and i², i^4 and made two equations.

 

  1. I – 3l + 5I – 7I……+ 1001I  

If we pair the two numbers starting from the beginning, there are exactly 250 pairs and they all equal -2I with extra +1001I.

So 250 X (-2I) = -500I and adding +1001I gives 501I

 

2) -2 + 4 -6 + 8……. + 1000

We can simplify this to understand more easily

2(-1 + 2 – 3 + 4 …. + 500)

We notice that there are 250 pairs which result of 1

2(250*1) = 500

So ∴ 500 + 501I

 

Easier way is

more answer

more answer 2

Referenced from

http://cs.smu.ca/~nsml/OldContests/2010Game2Problems.pdf

http://cs.smu.ca/~nsml/OldContests/2010Game2Solutions.pdf

 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s