Sum of GP Formula

In our previous article, we provided a brief introduction to Geometric Progression (i.e., GP) and the GP formula. In this article, we will focus on how to calculate the sum of a given GP. We will discuss the formula for finding the sum of both finite and infinite GPs, and we will demonstrate this concept with the help of various examples.

Table of Content

• What is Geometric Progression?
• Sum of Finite Geometric Progression: Derivation and Formula
• Examples
• Sum of Infinite Geometric Progression: Derivation and Formula
• Examples
• Special Cases and Consideration for Geometric Progression
• When the Common Ratio is Greater than 1
• When the Common Ratio is Between -1 and 1
• When the Common Ratio is Less than -1

What is Geometric Progression?

A geometric progression (GP), also known as a geometric sequence, is a special kind of sequence where each term is obtained by multiplying the previous term by a constant number called the common ratio. 

Example:

• 10, 20, 40, 80, 160, ... (common ratio 2)
• 1/2, 1/4, 1/8, 1/16, ... (common ratio 1/2)

Note: 

• The formula to find the n-th term of GP is a*r^(n-1), where
• a: first term
• r: common ratio
• Types of Geometric Progression
• Finite Geometric Progression
• Infinite Geometric Progression

Now, let's learn how to find the sum of any given GP, following our clear understanding of what GP is.

Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric progression can be calculated using two different formulas, depending on whether the common ratio (r) equals 1 or not.

Case-1: When r is not equal to 1

The sum of the first n terms (Sn) is given by:

Sn = a1*(1-r^n) / 1 - r, 

where

a1 is the first term, r is the common ratio, and n is the number of terms 

Case-2: when r = 1

In this case, the sum of the first n terms is simply given by:

Sn = n*a_1

Let's prove the above formula.

Derivation of Sum of the First n Terms of a Geometric Progression

Consider a geometric progression with the first term a and the common ratio r, and let Sn represent the sum of the first n terms. The progression can be written as

a, ar, ar^2, ar^3, ..., ar^(n-1)

Then the sum of these n terms is:

Sn = a + ar + ar^2 + ar^3 + ... + ar^(n-1).......(1)

Multiply both sides of the equation by r on both sides.

r*Sn = ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n....(2)

Now, subtracting the first from the second, we will get:

r*Sn - Sn = ar^n - a

=> Sn(r - 1) = a(r^n - 1)

=> Sn = a(r^n - 1)/(r - 1)

Hence, Proved.

Now, let's have some examples to master the formula.

Examples of the Sum of Finite Geometric Progression

Example-1: Basic GP Sum Calculation

Given: a = 2, r = 3, and n = 4.

Calculate the sum of the first 4 terms.

Solution:

Here, we will use the formula of the sum of the first n-terms of GP

S4 = 2*(3^4 - 1)/(3-1) = 2(81 - 1)/2 = 80.

Example-2: GP with Fractional Common Ratio

Given: a = 5, r = 1/2, n = 3

Calculate the sum of the first 3 terms.

Solution:

Using the sum of the first n terms in sequence.

S3 = 5*((1/2)^3 - 1) / ((1/2) - 1) = 35/4 = 8.75

=> S3 = 8.75

Example-3: GP with Negative Common Ratio

Given: a = 1, r = -2, and n = 3

Calculate the sum of the first 3 terms.

Solution:

S3 = 1*((-2)^3 - 1)/(-2-1) = 1*(-8-1)/-3 = 3

Example-4: Large n Value in GP

Given: a = 1, r = 2, and n = 10

Calculate the sum of the first 10 terms.

Solution:

S10 = 1*(2^10 - 1) / (2-1) = 1023

=> S10 = 1023

Sum of Infinite GP Formula

An infinite GP consists of an endless sequence of terms, where each term is derived by multiplying the previous terms by a constant called the constant ratio. Not all infinite GPs have a finite sum, but under some circumstances, these infinite GPs converge to a specific value.

Condition for Convergence:

Any infinite GP converges to a finite sum if the absolute value of the common ratio (r) is less than 1, i.e., |r| < 1. Otherwise, it will diverge.

The sum of an infinite GP is a/1-r.

Now, let's prove this.

S_inf = a + ar + ar^2 + ar^3 + ........ (1)

Multiply both sides by r gives:

r*S_inf = ar + ar^2 + ar^3 + ar^4 + ......... (2)

Now, subtracting (2) from (1), we will get:

S_inf - r*S_inf = a

=> S_inf(1-r) = a

=> S_inf = a/(1-r)

Hence, Proved.

Now, let’s have some examples to ace the concepts.

Examples of the Sum of Infinite Geometric Progression

Example-1: Negative Common Ratio

Given: a = 10, and r = -1/4

Calculate the sum of the series.

Solution:

S_inf = 10 / 1 + (1/4) = 40/5 = 8

=> S_inf = 8

Example-2

Given: a = 100, and r = 0.99

Calculate the sum of a series with a common ratio close to 1.

S_inf = 100 / 1 - 0.99 = 10000

=> S_inf = 10,00 

Special Cases and Consideration for Geometric Progression

When the Common Ratio is Greater than 1

• Behavior: The GP exhibits exponential growth. Each term is larger than the previous one, leading to rapidly increasing values.
• Implication: Such sequences are used to model scenarios of exponential increase, like population growth under ideal conditions or investment growth with a constant high return rate.
• Consideration: For infinite series with r > 1, the sum diverges to infinity, meaning it grows without bounds and does not converge to a finite sum.

When the Common Ratio is Between -1 and 1

• Behavior: The GP may exhibit exponential decay (0 < r < 1) or oscillating behavior (−1 < r < 0). For positive r, each term decreases in value, approaching zero. For negative r, the terms alternate in sign, leading to a series that oscillates in value.
• Implication: Sequences with a positive r in this range are often used to model depreciation or cooling processes. Sequences with negative r can model scenarios where values alternate in direction, such as alternating current in electrical engineering.
• Consideration: Infinite series with ∣r∣ < 1 converge to a finite sum. The formula S∞​= a/1−r​ applies, allowing for calculating the sum of all terms in the series.

When the Common Ratio is Less than -1

• Behavior: The GP exhibits divergent behavior, with terms increasing in magnitude but alternating in sign. This leads to a sequence that neither converges to a finite value nor stabilizes around a single value.
• Implication: Such sequences are less common in practical applications due to their divergent nature. However, they might be used in theoretical studies or mathematical explorations of series behavior.
• Consideration: For r ≤ −1, the series does not have a sum in the traditional sense, as the terms do not approach a finite limit but instead become infinitely large in magnitude.

Conclusion

In conclusion, the sum of the GP formula is essential for finding the sum of a GP, whether it's the sum of the first n terms or the sum of an infinite GP. When solving word problems, identifying keywords can be helpful, but it's important also to develop a deeper understanding of the underlying mathematical concepts.

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