How to Find the Sum of a Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In a geometric series, if the absolute value of the common ratio (∣r∣) is less than 1, the series converges to a finite value. Otherwise, it diverges (grows without bound). In this article, we will discuss the formula for finding the sum of terms of a geometric series.

What is a Geometric Series?

A geometric series is a sequence of numbers in which each term except the first is obtained by multiplying the preceding term with a constant value known as a common ratio.

The common ratio can be less than 1, equal to 1 or greater than 1. A geometric series is also known as a geometric progression, abbreviated as GP. Mathematically, it is represented as follows:

a, ar, ar², ar³, . . ., ar^n-1 (n terms)

Where,

• a is the first term,
• r is the common ratio, and
• n is the number of terms.

Examples of Geometric Series

Some examples of geometric series are:

• 2, 4, 8, 16, 32, 64, 128, . . . (Common Ratio = 2)
• 9, 3, 1, 1/3, 1/9, 1/27, . . . (Common Ratio = 1/3)
• 4, 4, 4, 4, 4, . . . (Common Ratio = 1)

The n^th term of a GP can be derived as follows:

a_n = ar^n-1

Sum of a Geometric Series

The sum of a geometric series up to a finite number of terms is derived as follows. Let us call S_n as the sum of a geometric series up to n terms. Then, we have,

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

⇒ S_n = a × (1 + r + r² + r³ + . . . + r^n-1)

We have a relation that,

1 – rⁿ = (1 – r) × (1 + r + r² + r³ + . . . + r^n-1)

Using this relation, we get,

S_n = a × (1 – rⁿ)/(1 – r)

Thus, we have derived the expression for sum of a geometric series. For convenience in use, formula for different common ratio of GP is written as follows:

For Common Ratio < 1

S_n = a × (1 – rⁿ)/(1 – r)

Where,

• S_n is sum of the GP upto n terms.
• a is the first term,
• r is the common ratio, and
• n is the number of terms up to which sum is required.

For Common Ratio > 1

S_n = a × (rⁿ – 1)/(r – 1)

Where,

• S_n is sum of the GP upto n terms.
• a is the first term,
• r is the common ratio, and
• n is the number of terms up to which sum is required.

Sum of an Infinite Geometric Series

The sum of a geometric series having common ratio less than 1 up to infinite terms can be found. Let us derive the expression for sum as follows. We have, sum of a geometric series up to n terms given by,

S_n = a × (1 – rⁿ)/(1 – r)

When, r<1 and n tends to infinity, rⁿ tends to zero. Thus, above expression takes the form,

S_∞ = a/(1-r)

Hence, above expression can be used to find sum of an infinite geometric progression having common ratio less than 1. Please note that above expression is valid only for geometric series having common ratio less than 1 and fails in case of common ratio being greater than 1.

Solved Examples on Sum of a Geometric Series

Example 1: Find the sum of first 5 terms of a geometric series having first term as 1 and common ratio as 2.

Solution:

We know that, sum upto n terms of a GP is given by,

⇒ S_n = a × (rⁿ – 1)/(r – 1)

Here, a = 1, r = 2, and n = 5,

Putting all these values in the formula for S_n, we get,

⇒ S_n = 1 × (2⁵ – 1)/(2 – 1) = 31

Thus, sum of the given series upto 5 terms is found to be 31.

Example 2: Find the sum of an infinite GP series having first term as 4 and common ratio as 1/2.

Solution:

We know that, sum of an infinite GP series is given by,

⇒ S = a/(1-r)

Here, a = 4 and r = 1/2, Putting these values in above expression,

⇒ S = 4/(1-1/2)

⇒ S = 4/(1/2) = 8

Thus, we get sum of an infinite GP series having first term as 4 and common ratio 1/2 as 8.

Practice Problems on Sum of a Geometric Series

Problem 1: Find sum of the series: 4, 12, 36, 108, . . . up to 6 terms.

Problem 2: Find sum of the infinite series: 64, 16, 4, 1, . . . up to infinite terms.

Problem 3: Find sum of a GP having first term a = 5 and common ratio = 2 up to 7 terms.

Problem 4: What would be sum of a GP series having a = 1 and common ratio = 1/2 up to 5 terms.

Problem 5: Determine the sum: 40 + 10 + 2.5 + up to infinite terms.

FAQs on Sum of a Geometric Series

Define geometric series.

A geometric series is a sequence of numbers in which ratio of each succeeding term to the preceding term remains the same. This ratio is called common ratio of the GP.

What is the formula for the sum of a geometric series?

The formula to find sum of a geometric series is given as, S = a × (rⁿ – 1)/(r – 1), where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

When can we use the formula for the sum of an infinite geometric series?

The formula for the sum of an infinite geometric series can be used when the absolute value of the common ratio ∣r∣ is less than 1. In this case, the sum is given by:

S = a/(1 – r)

What happens if the common ratio r is greater than or equal to 1?

If the absolute value of the common ratio ∣r∣ is greater than or equal to 1, the series diverges, meaning it grows without bound as the number of terms increases.

What are some applications of geometric series in real life?

Geometric series have various applications in fields such as finance (e.g., compound interest calculations), population growth models, computer algorithms, and physics (e.g., modeling radioactive decay).