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:
“Much of our American progress has been the product of the individual who had an idea; pursued it; fashioned it; tenaciously clung to it against all odds; and then produced it, sold it, and profited from it.”
—Hubert H. Humphrey (1911–1978)
“The guys who fear becoming fathers don’t understand that fathering is not something perfect men do, but something that perfects the man. The end product of child raising is not the child but the parent.”
—Frank Pittman (20th century)
“[The political mind] is a strange mixture of vanity and timidity, of an obsequious attitude at one time and a delusion of grandeur at another time. The political mind is the product of men in public life who have been twice spoiled. They have been spoiled with praise and they have been spoiled with abuse.”
—Calvin Coolidge (1872–1933)