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Einsein had once said that the strongest power in this world is the power of compounding.

The story (not by Einstein) goes as: When a great mathematician developed the game of chess, he brought it to the court of the king and explained the game. The king as well as the courtiers were amazed and the king promised him any thing on the earth, he wished, as the reward. The mathematician said, I am a poor man and I have no great wishes, but since if you want to reward me, just put one paisa (Indian cent) on the first square, 2 paisa on the second, and so on, go on doubling the amount of money on every square till the 64th one. The king said, I pity on you great man, you could have asked for much more. Anyway, keep this bag of 10,000 rupees, arrange the petty paisa on the squares the way you like and keep the balance money. The mathematician insisted that this operation be done by the king's men. Anyway, the king agreed and ordered his treasurer to do the needful.

After sometime, the treasurer came to the king and said, Your highness, 10,000 rpees are all gone and still squares are left over. The king, a bit surprised, told him to use whatever extra money is required and not to disturb him for these small issues. After some more time, the treasurer again came running, in a nervous state, and said, excuse me Your highness, but our treasury is exhausted and still squares are left over. The king was aghast - we have billions of billions of rupees in our treasury and you mean all of it is exhausted by putting some petty paisa on the squares. Yes my lord ...

Well, the problem is as follows:

If you deposit $100 in a bank on simple interest rate of 10% per anum, the money will be doubled in 10 years. If another bank offers compound interest - compounded anually, it will be doubled in much less time. Better still, if it is coumpounded monthly. Find out the absolute minimum time in which it will double. No approximation please.

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any idea for the 3rd one (problem #4)?

if we look the problem from a mathematic viewpoint we have an alternating series where we can arrange the numbers in any desired order, in our case (conditionally convergent series) this means that with rearranging the numbers we can get a convergent or a divergent series as well, moreover we can reach any number, thats why I said you have found two solutions, cause every number is a solution,

so if you add up the same numbers you can get any value, kinda weird, but true, the intuitions are not always working in the realm of infinity

more info: conditional convergence

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