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:
“Perhaps I am still very much of an American. That is to say, naïve, optimistic, gullible.... In the eyes of a European, what am I but an American to the core, an American who exposes his Americanism like a sore. Like it or not, I am a product of this land of plenty, a believer in superabundance, a believer in miracles.”
—Henry Miller (1891–1980)
“Humour is the describing the ludicrous as it is in itself; wit is the exposing it, by comparing or contrasting it with something else. Humour is, as it were, the growth of nature and accident; wit is the product of art and fancy.”
—William Hazlitt (1778–1830)
“The UN is not just a product of do-gooders. It is harshly real. The day will come when men will see the UN and what it means clearly. Everything will be all right—you know when? When people, just people, stop thinking of the United Nations as a weird Picasso abstraction, and see it as a drawing they made themselves.”
—Dag Hammarskjöld (1905–1961)