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Let be a sequence of real numbers. Compute:
First and foremost we set and it is obvious that . We are making use of probabilistic methods. Suppose than an infinite number of coins are flipped. Let be the probability that the -th coin toss lands heads and let us consider the first time heads comes up. Then is the probability that the first head appears in the – th flip and is the probability that all flips come up tails. Thus,
Prove that in any triangle it holds that
where denotes the circumradius and the inradius.
Using the law of sines we have that
and if we denote the area of the triangle then
Solve the system
We set and where . The second equation of course is written as
Now ‘s follow.
Historical note: In the Babylonian signs , tracing back in 1700 BC , there are a lot of geometrical problems that are equivalent to the solution of such systems. In order to be solved the following ( modern ) formulae were used
The above procedure actually led to the discriminant in order for us to solve a second order equation. For example in a book of that age we see the equation . Try to solve this using the above technique.
The above exercise , along with the historical note , can be found at mathematica.gr .
Evaluate the product
Then noting that we have that
Consider the harmonic sequence
Prove that if we pick dinstinct terms of the above sequence we can construct an arithmetic progression sequence of as large (finite) length as we want.
are all distinct terms of the sequence.