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:
“The end product of child raising is not only the child but the parents, who get to go through each stage of human development from the other side, and get to relive the experiences that shaped them, and get to rethink everything their parents taught them. The get, in effect, to reraise themselves and become their own person.”
—Frank Pittman (20th century)
“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)
“Everything that is beautiful and noble is the product of reason and calculation.”
—Charles Baudelaire (1821–1867)