Geometric Sum Formula

About Geometric Sum Formula

Let us first define a geometric sequence before learning the geometric sum formula. A geometric sequence is one in which each phrase has a constant ratio to the one before it. A geometric sequence with a limited number of terms and the initial term a and the common ratio r is often written as
a, ar, ar2, ..., arn-1

The sum of terms in a geometric sequence is called a geometric sum. The sum of terms in the geometric sequence is calculated using the geometric sum formula.

The geometric sum formula is a formula for calculating the total of all terms in a geometric sequence. Two geometric sum formulas exist. The first is used to discover the sum of a geometric sequence's first n terms, while the second is used to find the sum of an infinite geometric series.

  • Geometric Sum Formula For Finite Terms
    Sum = a(1 - rn)/1 - r
  • Geometric Sum Formula For Infinite Terms
    Sum = a/1 - r

Geometric Sum Formula

  1. Geometric sum formula for finite terms :-
    If r = 1, Sn = na
    If |r| < 1 , Sn=a(1−rn)/1−r
    If |r| >1,Sn=a(rn−1)r−1
  2. Geometric sum formula for infinite terms :-
    If |r| < 1,S∞=a/1−r
    If |r| >1, Series does not converge & it has no sum.

Where,

  • a - first term
  • r - common ratio
  • n - number of terms

Derivation of Geometric Sum Formula

Sum of a geometric series Sn, with common ratio r
Sn = ∑i=1n ai = a(1 - rn/1 - r)

We will use the polynomial long division formula.

  • Sum of first n terms of Geometric progression is
    Sn =a + ar + ar2 + ar3 + ... +arn–2 + arn–1.....(1)
  • Multiplying both sides by r, we get
    rSn =ar + ar2 + ar3+ ... + arn–2+ arn–1+ arn ....(2)
  • (2)−(1) gives rSnSn -SnSn = arn- a ⇒ Sn(r-1) = a(rn- 1)
  • Hence we have derive the sum of n terms of GP as Sn=a(rn−1)/r−1

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Geometric Sum Formula
Geometric Sum Formula

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