Geometric Series (original) (raw)

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 \sum_{k=1}^{\infty}a_k {Infinite Series}
a + ar + ar2 + ar3 + ar4 + . . . arn OR \sum_{k=1}^{n}a_k{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,

\bold{{S_n =\frac{a(1-r^n)}{1-r}}}

Where

Sn = sum up to nth term,

a = First term, and

r = common factor.

**Derivation for Geometric Series Formula

Suppose a Geometric Series for n terms:

Sn = a + ar + ar2 + ar3 + .... + arn-1 . . . (1)

Multiplying both sides by the common factor (r):

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

Subtracting Equation (1) from Equation (2):

(r Sn - Sn) = (ar + ar2 + ar3 + ar4 +. . . arn) - (a + ar + ar2 + ar3 + . . . + arn-1)

⇒ Sn (r-1) = arn - a

⇒ Sn (1 - r) = a (1-rn)

⇒ {S_n =\frac{a(1-r^n)}{1-r}}

**Note: When the value of k starts from 'm', the formula will change.

\sum_{k=m}^{n}ar^k=\frac{a(r^m-r^{n+1}}{1-r}, when r≠0

**For Infinite Geometric Series

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

S_\infty = \sum_{n=1}^{\infty}ar^{n-1} = \frac{a}{1-r}; -1<{r}<1

nth term for the G.P. : an = arn-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, ar 2 , ar 3 , ar 4 , . . .	a + ar + ar2 + ar3 + ar4 + . . .
**Example	2, 6, 18, 54, . . .	2 + 6 + 18 + 54 + . . .

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