Geometric Series

Last Updated : 23 Jul, 2025

In a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease.

Geometric Sequence is given as:

a, ar, ar², ar³, ar⁴,..... {Infinite Sequence}
a, ar, ar², ar³, ar⁴, ....... arⁿ {Finite Sequence}

Geometric Series for the above is written as:

a + ar + ar² + ar³ + ar⁴ + . . . OR {Infinite Series}
a + ar + ar² + ar³ + ar⁴ + . . . arⁿ OR {Finite Series}

Where,

a = First term, and r = Common Factor.

Convergence of Geometric Series

The convergence of a geometric series (infinite) depends solely on the value of the common ratio r:

Convergent Series: The series converges if the absolute value of the common ratio is less than 1: ∣r∣ < 1
Divergent Series: The series diverges if the absolute value of the common ratio is equal to or greater than 1: ∣r∣ ≥ 1

Geometric Series Formula

The Geometric Series formula for the Finite series is given as,

Where
S_n = sum up to n^th term,
a = First term, and
r = common factor.

Derivation for Geometric Series Formula

Suppose a Geometric Series for n terms:

S_n = a + ar + ar² + ar³ + .... + ar^n-1 . . . (1)

Multiplying both sides by the common factor (r):

r S_n = ar + ar² + ar³ + ar⁴ + ... + arⁿ . . . (2)

Subtracting Equation (1) from Equation (2):

(r S_n - S_n) = (ar + ar² + ar³ + ar⁴ +. . . arⁿ) - (a + ar + ar² + ar³ + . . . + ar^n-1)

⇒ S_n (r-1) = arⁿ - a

⇒ S_n (1 - r) = a (1-rⁿ)

⇒

Note: When the value of k starts from 'm', the formula will change.
, when r≠0

For Infinite Geometric Series

n will tend to Infinity, n ⇢ ∞, Putting this in the generalized formula:

n^th term for the G.P. : a_n = ar^n-1

Geometric Sequence Vs Geometric Series

Some of the common differences between Geometric Sequences and Series are listed in the following table:

Aspect	Geometric Sequence	Geometric Series
Definition	A sequence of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero number (common ratio).	The sum of terms in a geometric sequence.
General Form	a, ar, ar², ar³, ar⁴, . . .	a + ar + ar² + ar³ + ar⁴ + . . .
Example	2, 6, 18, 54, . . .	2 + 6 + 18 + 54 + . . .