Invention of Arithmetic Progression

Arithmetic Progression was invented by Johann Carl Friedrich Gauss. Here is the story how he found the method.

One day at school, Gauss’s teacher wanted to take a rest and asked the students to sum the integers from 1 to 100 as busy work. To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away.His classmates and teacher were astonished, and Gauss ended up being the only pupil to calculate the correct answer.

Here is how he calculated :

First, he wrote the sum twice, one in an ordinary order and the other in a reverse order:

1 + 2 + 3 + 4 + . . . + 99 + 100
100 + 99 + . . . + 4 + 3 + 2 + 1
By adding vertically, each pair of numbers adds up to 101:

1 + 2 + 3 + . . . + 98 + 99 + 100
100 + 99 + 98 + . . . + 3 + 2 + 1

101 + 101 + 101 + . . . + 101 + 101 + 101

Since there are 100 of these sums of 101, the total is 100Image101 = 10,100. Because this sum 10,100 is twice the sum of the numbers 1 through 100, we have:

1 + 2 + 3 + . . . + 98 + 99 + 100 = 100Image101 / 2 = 5050.
 
Using Gauss´ approach, we can easily derived the formula of Arithmetic Progression.

References:
http://blog.brilliant.org/2013/03/03/arithmetic-progression-and-young-gauss-the-prince-of-mathematics/
http://www.education2000.com/demo/demo/botchtml/arithser.htm
http://www.jimloy.com/algebra/gauss.htm

The Twin Paradox and Time Travel


The Twin Paradox..
Einstein came up with an example to show the effects of time dilation that he called the "twin paradox."
Let's consider a pair of twins, Al and Bert, both of whom are 10 years old in their highly futuristic universe.

Al's parents decide to send him to summer camp in the Alpha-3 star system, which is 25 light-years away (a light-year is the distance light travels in a year).
Bert doesn't want to go and stays home on Earth. So Al sets out on his own.
Wanting him to get there as quickly as possible, his parents pay extra and send him at 99.99 percent the speed of light.
The trip to the star and back takes 50 years. What happens when Al returns? His twin brother is now 60 years old,
but Al is only 10 and a half. How can this be? Al was away for 50 years but only aged by half a year. Has Al just discovered the fountain of youth?

Not at all. Al's trip into space lasted only a half year for him, but on Earth 50 years passed.
Does this mean that Al can live forever? Nope. He may have aged by only half a year in the time it took 50 years to pass on Earth,
but he also only lived half a year. And since time can slow down but never goes backwards, there's no way he could grow younger.

The time difference is calculated using the following Time dilation equation:
                                
                     tR = tM/(1 - (v2/c2))1/2

Therefore, if a person travels at the speed of 99.99 % of the speed of light for 10 years, and then returns to earth.The time elapsed on earth will be 1000 years.In a way he travelled to the future.

References:
http://www.pbs.org/wgbh/nova/einstein/hotsciencetwin/
http://mitchellscience.com/twin_paradox_calculations