Product
The product of a geometric progression is the product of all terms. If all terms are positive, then it can be quickly computed by taking the geometric mean of the progression's first and last term, and raising that mean to the power given by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence: take the arithmetic mean of the first and last term and multiply with the number of terms.)
- (if ).
Proof:
Let the product be represented by P:
- .
Now, carrying out the multiplications, we conclude that
- .
Applying the sum of arithmetic series, the expression will yield
- .
- .
We raise both sides to the second power:
- .
Consequently
- and
- ,
which concludes the proof.
Read more about this topic: Geometric Progression
Famous quotes containing the word product:
“He was the product of an English public school and university. He was, moreover, a modern product of those seats of athletic exercise. He had little education and highly developed muscles—that is to say, he was no scholar, but essentially a gentleman.”
—H. Seton Merriman (1862–1903)
“The history is always the same the product is always different and the history interests more than the product. More, that is, more. Yes. But if the product was not different the history which is the same would not be more interesting.”
—Gertrude Stein (1874–1946)
“...In the past, as now, [Hollywood] was a stamping ground for tastelessness, violence, and hyperbole, but once upon a time it turned out a product which sweetened the flavor of life all over the world.”
—Anita Loos (1888–1981)