Is there any trick to solve it in less than 15 sec ???

Warning: this 10-second trick depends on having an idea what the complex plane is and how operations on it map to geometric transformations - very useful stuff to know. If you invest a little effort in this, you can also learn or create many similar tricks useful even on much more difficult problems.

Short version of trick:

As with almost all summation problems, start with just the first ('last' can work also) iteration of the terms (set n=1 here) and get a feel for the first iteration, then the second, then imagine extending this for all the steps in the entire sum. You can do this in terms of algebraic identities and so on, but at least for me, I go several times faster if I make /pictures/ in my mind and 'see' the problem. You specifically asked for a trick that works in less than 15 seconds, and when time is a constraint like that, I find that visualizing is a lot faster than dredging up facts and applying formulae.

In this case, first picture the complex plane. (a quick trip to Wikipedia can teach you how to do this.) The first (/n=1/ )iteration of the terms gives us /i^1 + i^2 = i + (-1) = i -1/ . /i/ is represented as a dot above the origin at 12 o'clock relative to the origin, and /i^2 = -1/ as a dot at the 9 o'clock position.

Now picture the second iteration. /i^2 + i^3 = -1 + (-i) = i -1/ . Immediately we see in our minds that the dots have rotated by 90 degrees counterclockwise to 9 and 6 o'clock positions respectively, and my mind at least leaps to the realization that on every subsequent iteration, the dots will rotate an additional 90 degrees, so they just march around in a circle by steps of a quarter-circle each time.

The two dots march around the circle in quarter steps three times by the twelfth iteration and you can just /see/ that the sum of all these contributions will be zero, by symmetry. This is very quick. Or see note 1 below.

Now that you see that the first 12 iterations cancel out, you are left with only the 13th, and you can see in your mind that the two dots are back in the same position they were at the first iteration, ie. /i/ and /-1/. So the total sum is the complex number /i - 1/.

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Note 1: complex numbers add much like two dimensional vectors, and if you are used to vector addition, you realize that when diametrically opposite numbers are added they sum to zero. The two terms in each iteration march around by 90 degrees at each step, so their position (on the map in our mind) is rotated by 180 degrees at every second step. That means that the contributions from iteration /n=1/ and iteration /n=3/ are opposite to each other, and the same goes for iterations 2 and 4. You can pair up iterations 1 and 3, 5 and 7, 9 and 11 in your mind and see they cancel, and the same for all even iterations. This is slower than just seeing the symmetry but some people like to think this way so I include this note.

Note 2:

I invented the particular 'trick' to solve this problem in about six to eight seconds, then implemented it in less than two seconds. The important thing here is not to learn this one specific trick designed to solve just this problem, but to pick up the general technique or /basic/ trick of tapping into your hard-won ability to visualize, turn the symbols into pictures, and then let the pictures evolve and look at what the pictures tell you.

In our culture, Rene Descartes famously connected algebra with pictures in this way and this led to immense progress. Many many problems can be visualized, but some visualizations are difficult or misleading so you have to be careful and, if you have time, check your answer by reproducing it with simple logical steps.

I find visualization of the problem to be a terrific time saver and a help in solving many otherwise intractable problems. However, it is in my mind very much a 'trick', a convenient way to draw on the human mind's enormous capability to imagine pictures quickly, and like all tricks, using it can be quite tricky.

Just practice the technique on problems, look and see how other people have done visualizations (there are a lot of good videos on the Web about this, as well as many books and static websites) and you can gradually learn when to use it and when to carefully use logic and symbols or thinking in words. Mathematicians use all kinds of thinking but this is one of the fastest, after "intuition" it might be the second fastest.

$i+2(i^2+...+i^{13})+i^{14}$ $i+2(3(i^2+i^3+i^4+i^5))+i^{14}$ $i+0+i^{14}$ $i-1$