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)
“A gentleman opposed to their enfranchisement once said to me, “Women have never produced anything of any value to the world.” I told him the chief product of the women had been the men, and left it to him to decide whether the product was of any value.”
—Anna Howard Shaw (1847–1919)
“Man’s main task in life is to give birth to himself, to become what he potentially is. The most important product of his effort is his own personality.”
—Erich Fromm (1900–1980)