What are Geometric Series?

1
What are Geometric Series?
2
Sissas Reward
If you take a chess board and place one piece of
rice on the first square, two on the second
square, four on the third square, eight on the
fourth square etc. How many pieces of rice would
you have in total on the whole chess board?
3
Consider n terms of a Geometric Sequence with
first term a and common ratio r a, ar, ar2,
ar3, ar4, , arn-2, arn-1 Take the sum of
the first n terms call this 1 Sn a ar
ar2 arn-1 Multiply the sum of the first
n terms by r, the common ratio call this 2
rSn ar ar2 arn-1 arn Subtract 2
from 1 Sn rSn a ar ar2 arn-1
(ar ar2 arn-1 arn) Group like terms
together Sn rSn a (ar ar) (ar2
ar2) (arn-1 arn-1) - arn Cancel out
any terms that will Sn rSn a - arn Take
out common factors Sn(1 r) a(1
rn) Rearrange to find the formula for the sum of
the first n terms
4
Sissas Reward
Sequence is 1, 2, 4, 8, 16, 32, a
r
2
1
Chessboard has 64 squares so n 64
Sn 18 446 744 073 709 551 615