**Geometric Progression**and denoted as

**G.P**Here will teach you about Geometric Sequences and Series.

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence i.e a sequence of numbers in which the ratio between consecutive terms is constant.

G.P = {a, ar, ar^{2}, ar^{3}, …ar^{(n-1)}, ar^{n} }

r = *ar*a = *ar ^{3}*ar

^{2}= …=

*ar*ar

^{n}^{(n-1)}

where:

a is the first term

r is the factor between the terms (called the “common ratio”)

Example: 1, 3, 9, 27, 81 is a geometric sequence. Here common ratio is 3.

nth term of a G.P

a_{n} = ar^{(n – 1)}

nth term from the last term

a_{n} = *1*r^{(n-1)}

where,

r is common ratio

a is first term

a_{n-1} is the term before the n th term

n is number of terms

**Sum of Terms in a G.P**

nth partial sum of a geometric sequence

Sum to infinity

where Sn is the sum of GP with n terms

S_{∞} is the sum of GP with infinitely many terms

a1 is the first term

r is a common ratio

n is the number of terms

**Problem 1**: Find the 10th term of below sequence

10, 30, 90, 270, 810, 2430, …

**Solution:** here n = 10 , r = 3

a_{n} = ar^{(n – 1)}

a_{10} = 10 x 3 ^{(10 – 1)}

a_{10} = 10 x 3 ^{(9)}

a_{10} = 10×19683 = 196830

**Problem 2**: Finf the sum of sequence If a1 = 1, r = 2 and n = 7.

S_{7} = *1 – (1 – 2 ^{7})*1 -2

S

_{7}=

*1 – 128*– 1

S

_{7}=

*– 127*– 1

S

_{7}= – 127

### >Geometric Mean

The Geometric Mean is a special type of average. The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, …, an, the geometric mean is defined as

G.M = n√(a1 × a2 × … × an)

**Problem 1**: What is the geometric mean of 4,8.3,9 and 17?

**Solution**: Here, no = 5;

As we know G.M = n√(a1 × a2 × … × an)

G.M = 5√(4 × 8 × 3 × 9 × 17) = (4 × 8 × 3 × 9 × 17)^{(1/5)} = 6.81

I hope, this article will help you a lot to understand the **Geometric Sequences and Series | Geometric Mean**. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.

**Related Topic:**

Sequence and Series | Progression | Type of Sequence

Arithmetic Sequences and Series | Arithmetic Mean

Harmonic Sequence | Harmonic Mean

Number Series Concept And Tricks | Problems