## Non linear system

Solve the system

Solution

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 .

## Finite trigonometric product

Evaluate the product

Solution

One good way to begin is by invoking the identity

Then noting that we have that

## Arithmetic – Harmonic Progression

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.

Solution

We just observe that

are all distinct terms of the sequence.

## A binomial squared sum

Prove that

Solution

Let us begin by recalling the binomial expansion

However,

Equatating the coefficients of we have that

yielding the result.

## A trigonometric sum

Evaluate

Solution

We note that

Thus summing telescopically we have that

since .