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:
“Junk is the ideal product ... the ultimate merchandise. No sales talk necessary. The client will crawl through a sewer and beg to buy.”
—William Burroughs (b. 1914)
“The seashore is a sort of neutral ground, a most advantageous point from which to contemplate this world. It is even a trivial place. The waves forever rolling to the land are too far-traveled and untamable to be familiar. Creeping along the endless beach amid the sun-squall and the foam, it occurs to us that we, too, are the product of sea-slime.”
—Henry David Thoreau (1817–1862)
“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)