Geometric sequences
A geometric sequence of numbers - click here - is characterized by the fact that each of the sequence members is created by multiplication with the constant factor q from the preceding member. Each member of the sequence (except the first) is the geometric mean of its two neighboring members.
Concerning the game of chess, which is known to be played on a board of 8⋅8=64 squares, there is the following anecdote:
ZETA, the inventor of the game, is said to have demanded from the emperor SHERAM a quantity of wheat as a reward - namely, one grain on the first square of the chessboard, two grains on the second square - https://domyhomework.club/algebra-homework/ , and on each additional square always twice the number of grains of the previous one.
This gives a total of 264-1 grains (which is about 1.84⋅1019 grains). If we now calculate 10 grains to one gram - https://domyhomework.club/analogy-homework/ , this gives about 9.2⋅1012 t of wheat. (The world harvest in 1994 was about 5.3⋅108 t, so it took more than ten thousand times the wheat harvested in 1994, so much has not been harvested in the world as a whole).
The example shows impressively that the sequence of the numbers 1; 2; 4; 8; 16 ... grows very fast.
A geometric sequence is characterized by the fact that the quotient q between two adjacent members is always equal, i.e., applies to all members of the sequence:
an+1an=q
Examples:
( 1 ) 2; 6; 18; 54; 162; 486 ... q=3
(2) 64; 48; 36; 27; 814; 24316 ... q=34
(3) 20; 2; 0,2; 0,02; 0,002; 0,0002 ... q=0,1
(4) 7; 7; 7; 7; 7; 7; 7 ... q=1
(5) -2; 2; -2; 2; -2; 2 ... q=-1
(6) 400; -200; 100; -50; 25; -12,5 ... q=- 0,5
By specifying the quotient q and the initial element a1, the whole sequence is determined, it is valid:
an=a1⋅qn-1
Useful Resources:
Development of fine dust pollution and countermeasures
Classical political liberalism
