How to Calculate the Sum to n Terms of a GP with Solved Examples
The concept of sum to n terms of a GP is essential in mathematics and helps in solving real-world and exam-level problems efficiently. It is commonly encountered in sequence and series questions across school, board, and competitive exams. Understanding this concept ensures you can quickly calculate the total of a geometric series with any number of terms.
Understanding Sum to n Terms of a GP
A sum to n terms of a GP (Geometric Progression) refers to the process of adding up the first n terms of a geometric sequence. A geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). This concept is widely used in compound interest calculations, population growth models, and physics problems involving repeated multiplications.
Formula Used in Sum to n Terms of a GP
The standard formula is: \( S_n = a\dfrac{(r^n - 1)}{(r-1)} \) if \( r \neq 1 \), where:
r = common ratio
n = number of terms
Here’s a helpful table to understand sum to n terms of a GP more clearly:
Sum to n Terms of a GP Table
| Type | Formula | Condition |
|---|---|---|
| General Case | \( S_n = a\dfrac{(r^n - 1)}{r - 1} \) | \( r \neq 1 \) |
| When r = 1 | \( S_n = n \times a \) | \( r = 1 \) |
| Alternative Form | \( S_n = a\dfrac{(1-r^n)}{1-r} \) | \( r \neq 1 \) |
This table shows how the pattern of sum to n terms of a GP changes based on the value of the common ratio.
Stepwise Proof of the Formula
1. Let the GP be \( a, ar, ar^2, \ldots, ar^{n-1} \), and let their sum be \( S_n \).
2. So, \( S_n = a + ar + ar^2 + \ldots + ar^{n-1} \).
3. Multiply both sides by r: \( rS_n = ar + ar^2 + ar^3 + \ldots + ar^n \).
4. Subtract the second equation from the first:
\( S_n - rS_n = a - ar^n \)
5. Factor out \( S_n \): \( S_n(1 - r) = a(1 - r^n) \).
6. Therefore, \( S_n = \dfrac{a(1 - r^n)}{1 - r} \), or equivalently
\( S_n = \dfrac{a(r^n - 1)}{r - 1} \) (by multiplying numerator and denominator by -1).
Worked Example – Solving a Problem
Example: Find the sum to first 5 terms of a GP with a = 2, r = 3.
1. Write the formula: \( S_n = a\dfrac{(r^n - 1)}{r-1} \).
2. Substitute the values: \( a = 2, r = 3, n = 5 \):
\( S_5 = 2\dfrac{(3^5 - 1)}{3-1} \)
3. Calculate \( 3^5 \): \( 3^5 = 243 \).
4. Compute numerator: \( 243 - 1 = 242 \).
\( S_5 = 2\dfrac{242}{2} \)
5. Simplify denominator: \( 3-1 = 2 \).
6. Calculate: \( S_5 = 2 \times 121 = 242 \).
Final Answer: The sum of the first 5 terms is 242.
Practice Problems
- Find the sum to first 7 terms of a GP where a = 4, r = 2.
- If the sum to n terms of a GP is 315, a = 5 and r = 2, find n.
- Calculate the sum of the first 6 terms for the GP: 3, 6, 12, ...
- A GP has a = 2, r = 0.5. Find the sum to the first 4 terms.
Common Mistakes to Avoid
- Using the AP (arithmetic progression) formula instead of the GP formula.
- Forgetting to check if r = 1, in which case use \( S_n = n \times a \).
- Not applying brackets properly for negative or fractional r.
Real-World Applications
The concept of sum to n terms of a geometric progression appears in areas such as calculating compound interest, spreading information through social networks, and modeling exponential growth and decay. Vedantu helps students see how these series form the foundation for advanced scientific and economic calculations.
Quick Comparison: AP vs GP vs HP
| Type | Sum to n Terms Formula |
|---|---|
| Arithmetic Progression (AP) | \( S_n = \frac{n}{2}[2a + (n-1)d] \) |
| Geometric Progression (GP) | \( S_n = a\frac{(r^n-1)}{r-1} \) |
| Harmonic Progression (HP) | Sum is taken by converting to AP of reciprocals |
We explored the idea of sum to n terms of a GP, how to apply it, how to solve problems step by step, and why it matters in real life. Practice more such questions on Vedantu and reinforce your understanding of geometric progressions and other sequences.
Explore Related Concepts
- Arithmetic Progression
- Sequence and Series
- Nth Term of AP
- Harmonic Progression
- Formula Sheet for Class 11
FAQs on Sum to n Terms of a Geometric Progression (GP) – Formula, Proof & Examples
1. What is the sum to n terms of a GP?
The sum to n terms of a GP (Geometric Progression) is the total obtained by adding the first n terms of a geometric sequence. It is calculated using the formula Sn = a[(r^n - 1)/(r - 1)] when r \u2260 1, where a is the first term and r is the common ratio. If r = 1, then Sn = n \u00d7 a.
2. What is the formula for sum to n terms of a geometric progression?
The sum of the first n terms of a geometric progression is given by:
• Sn = a[(r^n - 1)/(r - 1)] if r \u2260 1
• Sn = n \u00d7 a if r = 1
Here, a is the first term, r is the common ratio, and n is the number of terms.
3. How do you calculate the sum of first n terms in a GP when r>1?
When the common ratio r > 1, the sum of the first n terms is calculated by the formula: Sn = a[(r^n - 1)/(r - 1)]. Here, the power r^n grows larger as n increases, making the sum grow quickly. Always substitute your a, r, and n values carefully to solve.
4. How can you find the nth term of a GP?
The nth term of a GP is given by the formula: Tn = a \u00d7 r^{n-1}, where a is the first term, r is the common ratio, and n is the term number. This formula helps you identify any term in the sequence.
5. What are examples of sum of n terms of a GP in board exams?
Board exams often include problems like:
• Finding the sum of n terms with given a and r
• Calculating sum when r > 1 or r < 1
• Solving word problems involving money or growth modeled by GP
• Application of formulas in competitive exams like JEE
Practicing examples with various a, r, and n helps solidify understanding.
6. Why isn’t the AP sum formula used for GP questions?
The AP sum formula applies only to Arithmetic Progressions where terms increase by a constant difference. In contrast, a Geometric Progression has terms multiplied by a constant ratio. Since their growth patterns differ, the GP sum formula is essential to correctly compute sums of geometric sequences.
7. What mistake do students make when r=1 in GP?
When r=1, the GP becomes a sequence of identical terms. Students often mistakenly apply the general sum formula, which leads to division by zero. Instead, the correct formula is Sn = n \u00d7 a because all terms are equal to a. Always check for this special case to avoid errors.
8. How can I quickly recall the GP sum formula in exams?
To quickly recall the sum to n terms formula for GP:
1. Remember the core variables: a (first term), r (common ratio), and n (number of terms).
2. Use the formula frame: Sn = a × (r^n - 1)/(r - 1) when r \u2260 1.
3. For r = 1, just apply Sn = n × a.
4. Practice writing and solving examples for fast recall before exams.
9. Why does the sum formula change for infinite GP series?
The sum of an infinite GP exists only if the common ratio satisfies |r| < 1. In this case, the sum converges and the formula is S = a / (1 - r). If |r| >= 1, the series diverges, and no finite sum exists. This condition ensures the terms become smaller, allowing the infinite sum to have a finite value.
10. What if the common ratio is negative or a fraction?
If the common ratio r is a negative number or a fraction:
• When |r| < 1, the terms alternate in sign (if negative) and decrease in magnitude, allowing the sum to converge for infinite series.
• The sum to n terms formula still applies, just carefully substitute the negative or fractional r.
• Be mindful of sign changes when calculating and simplifying terms.
11. Why does the proof of the sum matter for viva questions?
Understanding the proof of the sum formula is important because:
• It demonstrates a clear grasp of how the formula is derived step-by-step.
• Helps in validating the formula's correctness.
• Answers viva questions that test conceptual knowledge rather than rote memorization.
• Strengthens problem-solving skills for advanced questions.
