### Geometric Sequences and Series | Geometric Mean

A geometric sequence is the type of sequence. Its also called 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, ar2, ar3, …ar(n-1), arn }

r = ara = ar3ar2 = …= arnar(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

an = ar(n – 1)

nth term from the last term

an = 1r(n-1)

where,
r is common ratio
a is first term
an-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
an = ar(n – 1)
a10 = 10 x 3 (10 – 1)
a10 = 10 x 3 (9)
a10 = 10×19683 = 196830

Problem 2: Finf the sum of sequence If a1 = 1, r = 2 and n = 7. S7 = 1 – (1 – 27)1 -2
S7 = 1 – 128– 1
S7 = – 127– 1
S7 = – 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

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