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, ..., ar^n-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, S_n = 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
S_n = $\sum _{i=1}^n$ a_i = 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 + ... +ar^n–2 + ar^n–1.....(1)
• Multiplying both sides by r, we get
  rSn =ar + ar² + ar³+ ... + ar^n–2+ ar^n–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

Get the list of all Maths formulas used in general calculations. 

Find Below pdf for downloading Geometric Sum Formula