General Geometric Sequence Formula

A geometric sequence, also known as a geometric progression, is a series of numbers where each term is obtained by multiplying the previous term by a fixed constant called the common ratio (r). The general formula for the nth term of a geometric sequence is given by:

R S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n

By multiplying both sides of this equation by r, we can rewrite it as:

S n = a 1 + a 1 r + a 1 r 2 + … + a 1 r n - 1

This formula allows us to calculate the sum of the first n terms of any geometric sequence. However, adding a large number of terms can be a tedious task.

For example, let's consider a geometric sequence defined by an = 3n+1, where n is the term number. The sum of the first five terms of this sequence is:

a 1 + a 2 + a 3 + a 4 + a 5

Using the formula above, we can calculate this sum as:

S 5 = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + a 1 r 4

Substituting the values of an and r, we get:

S 5 = (3(1)+1) + (3(1)+1)(3) + (3(1)+1)(3)^2 + (3(1)+1)(3)^3 + (3(1)+1)(3)^4

After simplifying the expression, we get:

S 5 = 34

This shows that the sum of the first five terms of this geometric sequence is indeed 34.

In general, given the first term a 1 and the common ratio r of a geometric sequence, we can write any term as:

an = a 1 r n-1

Using this formula, we can calculate the nth term of any geometric sequence. For example, if a 1 = 9 and the ratio between any two successive terms is 3, then the nth term of this sequence is given by:

an = 9(3)^n-1

This shows that geometric sequences are a powerful tool for modeling real-world phenomena where each term grows exponentially.

In conclusion, the general formula for the nth term of a geometric sequence is given by R S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n. This formula allows us to calculate the sum of the first n terms of any geometric sequence, making it a useful tool for modeling real-world phenomena where each term grows exponentially.

General geometric sequence formula