How to Calculate the Sum of a Geometric Progression with Formula & Examples
The concept of Geometric Progression Sum of GP plays a key role in mathematics and is widely applicable to topics like sequences, finance, population studies, and competitive exams. Knowing how to find the sum of a GP helps students solve problems quickly and accurately, both in school and in real life.
What Is Geometric Progression Sum of GP?
A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. For example, in the sequence 2, 4, 8, 16, each term is twice the one before (common ratio r = 2). The sum of GP means adding up terms of such a sequence. You’ll find GP sum applied in compound interest problems, computer algorithms, and modeling of real-world repeated multiplication situations.
Key Formula for Geometric Progression Sum of GP
Here’s the standard formula to find the sum of the first n terms in a geometric progression:
If the first term is \(a\) and common ratio is \(r\):
When \( r \neq 1 \):
\( S_n = a \dfrac{1 - r^n}{1 - r} \)
For an infinite GP (\( |r| < 1 \)), sum is:
\( S_{\infty} = \dfrac{a}{1 - r} \)
Cross-Disciplinary Usage
Geometric Progression Sum of GP is not only useful in Maths but also plays an important role in Physics (radioactive decay, oscillations), Computer Science (algorithms, data structures), and daily logical reasoning. JEE, NEET, and board exams often feature problems based on GP sum calculations. Many real-life patterns, like growth of populations or money doubling, follow this principle.
Step-by-Step Illustration
- Given the GP: 3, 6, 12, 24, ...
First term, \( a = 3 \), common ratio, \( r = 2 \) - Find the sum of the first 5 terms:
- Apply the formula:
\( S_5 = 3 \dfrac{1 - 2^5}{1 - 2} \) - Calculate \(2^5 = 32\):
\( S_5 = 3 \dfrac{1 - 32}{-1} = 3 \dfrac{-31}{-1} = 3 \times 31 = 93 \) - Final Answer: The sum of the first five terms is 93.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for mental calculation when the ratio is 2 and you want the sum of terms doubling each time (as in 2, 4, 8...):
Shortcut: Instead of adding each term, just use \( S_n = first\ term \times (2^n - 1) \). For example, for the sequence 5, 10, 20…, sum of 4 terms:
- \( 5 \times (2^4 - 1) = 5 \times (16 - 1) = 5 \times 15 = 75 \)
Speed tricks like this help save time in maths competitions and exams. Vedantu’s live teaching sessions include more such shortcuts to help you learn faster.
Try These Yourself
- Find the sum of the first 6 terms of the GP: 2, 6, 18, 54...
- Calculate the sum to infinity for the sequence 8, 4, 2, 1, ...
- What is the 7th term of the GP: 3, 9, 27, ...?
- Does the GP 1, -2, 4, -8, ... have a sum to infinity? Why or why not?
Frequent Errors and Misunderstandings
- Plugging in the wrong value for n (number of terms) in the formula.
- Applying the infinite sum formula when \( |r| \geq 1 \) (which is incorrect — only use when the ratio is between -1 and 1).
- Subtracting in the wrong order in \((1 - r^n)\) or the denominator \((1 - r)\).
- Confusing geometric progression (GP) with arithmetic progression (AP).
Relation to Other Concepts
The idea of Geometric Progression Sum of GP connects closely with topics such as Arithmetic Progression (where you add a constant) and Sequences and Series in general. GP and AP are regularly compared in exams, so understanding the differences is important for higher-level problem-solving.
Classroom Tip
Remember: For the infinite sum of a GP, check the absolute value of the ratio first. If \( |r| < 1 \), use the infinity formula. If not, you cannot sum it to infinity! Vedantu teachers often show this visually using graphs or blocks, making it much easier to grasp during live classes.
We explored Geometric Progression Sum of GP—from definition, formula, step-by-step examples, frequent mistakes, and valuable connections to other math ideas. Keep practicing with Vedantu to build confidence in using this formula for exams and real-world maths.
Direct Links to Related Topics
- Geometric Progression - Full Concept
- Nth Term of GP (Find Any Term)
- Sequences and Series - Revision
- Arithmetic Progression - AP vs GP
- Infinite Series and Convergence
FAQs on Geometric Progression: Sum of GP (Finite & Infinite)
1. What is a geometric progression (GP) in mathematics?
A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, 2, 4, 8, 16... is a GP with a common ratio of 2.
2. How do you calculate the sum of a finite geometric progression?
The sum of the first n terms of a GP (denoted Sn) is calculated using the formula: Sn = a(1 - rn) / (1 - r), where a is the first term, r is the common ratio, and r ≠ 1. If r = 1, then Sn = na.
3. What is the formula for the sum of an infinite geometric progression?
The sum of an infinite geometric progression (S∞) exists only if the absolute value of the common ratio |r| < 1. The formula is: S∞ = a / (1 - r), where a is the first term and r is the common ratio.
4. How do you find the common ratio (r) in a geometric progression?
To find the common ratio, divide any term by the preceding term. For example, in the GP 3, 6, 12, 24..., the common ratio is 6/3 = 2 (or 12/6 = 2, etc.).
5. What is the nth term of a geometric progression?
The nth term (an) of a geometric progression is given by the formula: an = arn-1, where a is the first term, r is the common ratio, and n is the term number.
6. What are some real-life applications of geometric progressions?
Geometric progressions model various real-world scenarios, including:
- Compound interest calculations: The growth of money invested with compound interest follows a geometric progression.
- Population growth (under certain conditions): If a population increases by a constant percentage each year, its growth can be modeled using a GP.
- Radioactive decay: The amount of a radioactive substance remaining after a certain time follows a geometric progression.
7. How do you solve word problems involving geometric progressions?
Word problems involving geometric progressions require identifying the first term (a), the common ratio (r), and the number of terms (n). Then, apply the appropriate formula (for the nth term or the sum of terms) to solve the problem.
8. What is the difference between an arithmetic progression (AP) and a geometric progression (GP)?
In an arithmetic progression, the difference between consecutive terms is constant. In a geometric progression, the ratio between consecutive terms is constant.
9. Why does the formula for the sum of an infinite geometric progression only work when |r| < 1?
When |r| ≥ 1, the terms of the geometric progression do not approach zero, and the sum of the infinite series will be infinite or undefined. The formula only converges to a finite sum when the terms get progressively smaller.
10. What happens if the common ratio (r) is equal to 1 in a geometric progression?
If r = 1, all terms in the geometric progression are equal to the first term (a). The sum of n terms is simply na.
11. Can a geometric progression have a negative common ratio?
Yes, a geometric progression can have a negative common ratio. The terms will alternate in sign (positive, negative, positive, etc.). The formulas for the sum and nth term still apply.
