Geometric Progression (GP): General term and sum A geometric progression (GP) is a sequence of numbers where the ratio between consecutive numbers is con...
A geometric progression (GP) is a sequence of numbers where the ratio between consecutive numbers is constant. This constant ratio is often represented by the letter 'r'. The first term of a GP is called the initial term, and the second term is called the common ratio.
General term:
The general term of a GP is given by the formula:
a_n = a_1 * r^(n - 1)
where:
a_n is the nth term of the GP
a_1 is the initial term
r is the common ratio
n is the position of the term in the sequence
Sum:
The sum of a GP can be found using the formula:
S_n = a_1 * (1 - r^n)
where:
S_n is the sum of the first n terms in the GP
a_1 is the initial term
r is the common ratio
Examples:
GP with r = 2:
a_1 = 10
a_2 = 20
a_3 = 40
a_4 = 80
a_5 = 160
These terms follow the pattern of adding 10 to the previous number. Therefore, the sum of the first 5 terms is:
S_5 = 10 * (1 - 2^5) = 100
GP with r = 0.5:
a_1 = 10
a_2 = 5
a_3 = 2.5
a_4 = 1.25
a_5 = 0.625
These terms follow the pattern of dividing the previous number by 2. Therefore, the sum of the first 5 terms is:
S_5 = 10 * (1 - 0.5^5) = 25
In conclusion:
A geometric progression is a sequence where the ratio between consecutive terms is constant. The general term of a GP is given by the formula a_n = a_1 * r^(n - 1), and the sum of the first n terms can be found using the formula S_n = a_1 * (1 - r^n)