How to Calculate the Sum of n Terms in Arithmetic Progression (AP)
The concept of sum of n terms plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing the formula for the sum of n terms helps you handle arithmetic sequences, geometric progressions, and even special series like the sum of natural numbers. Mastering this can boost both your calculation speed and accuracy, making it super useful for Board exams and competitive tests.
What Is Sum of n Terms?
A sum of n terms is the process or formula used to find the total of the first 'n' numbers in a sequence. You’ll find this concept applied in areas such as arithmetic progression (AP), geometric progression (GP), and summation of natural numbers or their squares and cubes. Whether you are solving series in maths, dealing with patterns in science, or handling computer algorithms, learning the sum of n terms formula is essential.
Key Formula for Sum of n Terms
Here’s the standard formula for an arithmetic progression (AP):
\( S_n = \dfrac{n}{2} [2a + (n - 1)d] \)
where a = first term, d = common difference.
If the nth (last) term is given:
\( S_n = \dfrac{n}{2} (a_1 + a_n) \)
For a geometric progression (GP):
\( S_n = a \dfrac{(r^n - 1)}{r - 1} \), where r ≠ 1
Cross-Disciplinary Usage
Sum of n terms is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. When solving problems involving repeated measurements, patterns, or even programming loops, this formula comes in handy. Students preparing for JEE or NEET will see its relevance in various questions on sequences and series.
Step-by-Step Illustration
Let’s see how to use the sum of n terms formula with an example:
1. Suppose the first term (a) = 5 and common difference (d) = 3. Find the sum of the first 10 terms of the AP.2. Formula: \( S_n = \dfrac{n}{2} [2a + (n-1)d] \ )
3. Substitute values: \( n = 10, a = 5, d = 3 \)
4. \( S_{10} = \dfrac{10}{2} [2 \times 5 + (10 - 1) \times 3] \)
5. \( S_{10} = 5 [10 + 27] = 5 [37] = 185 \)
Answer: The sum of the first 10 terms is 185.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for AP sums when you know both the first and last terms:
Formula: \( S_n = \dfrac{n}{2} (first\ term + last\ term) \)
Example Trick: Find the sum of the first 100 natural numbers.
Shortcut: Use n(n+1)/2
So, 100 × 101 / 2 = 5050.
Tricks like this help students solve questions faster in exams and are taught by Vedantu’s expert tutors in live classes.
Try These Yourself
- Write the sum of the first 15 terms of the AP: 4, 7, 10, ...
- Find the sum of the first 5 terms of the sequence: 3, 6, 12, 24, ... (GP)
- Calculate the sum of squares of the first 6 natural numbers.
- If the 12th term of an AP is 50 and the first term is 8, what is the common difference?
Frequent Errors and Misunderstandings
- Forgetting to use n/2 and multiplying by n instead.
- Confusing the formula between AP and GP sums.
- Using 2a + (n-1)d wrongly—parentheses matter!
- Not checking if the common difference (d) is positive or negative.
- Applying the AP sum formula to non-AP series.
Relation to Other Concepts
The idea of sum of n terms connects closely with topics such as sequence and series and arithmetic mean. Mastering this helps with understanding future concepts like infinite series and advanced algebra.
Classroom Tip
A quick way to remember the sum of n terms in AP: Pair first and last, second and second last, and so on—the sum of each pair is the same! Multiply by the number of pairs to get your answer. Vedantu’s teachers often draw arrows between paired terms to make this idea really stick during live classes.
We explored sum of n terms—from definition, formula, practical steps, common mistakes, and related maths concepts. Keep practicing with Vedantu’s online resources and problem sets to become more confident in working with sequence sums for all your exams and assignments!
| Type of Series | Sum of n Terms Formula | Example |
|---|---|---|
| Arithmetic Progression (AP) | \( S_n = \dfrac{n}{2}[2a + (n-1)d] \) | 2, 5, 8, ... |
| AP (with nth term known) | \( S_n = \dfrac{n}{2}(a_1 + a_n) \) | First = 7, 10th = 34 |
| Geometric Progression (GP) | \( S_n = a \dfrac{(r^n - 1)}{r - 1} \) | 3, 6, 12, ... |
| First n natural numbers | \( S_n = \dfrac{n(n+1)}{2} \) | 1 + 2 + 3 + ... + n |
| Sum of squares | \( S_n = \dfrac{n(n+1)(2n+1)}{6} \) | 1² + 2² + ... + n² |
| Sum of cubes | \( S_n = [ \dfrac{n(n+1)}{2} ]^2 \) | 1³ + 2³ + ... + n³ |
To level up your understanding, don’t miss our Proofs of Important Formulae resource for a stepwise breakdown of derivations.
FAQs on Sum of n Terms: Formulas, Examples & Applications
1. What is the formula for the sum of n terms of an arithmetic progression (AP)?
The sum of n terms of an arithmetic progression (AP), denoted as Sn, is given by the formula:
Sn = n/2 [2a + (n - 1)d]
where:
• a represents the first term of the AP
• d represents the common difference between consecutive terms
• n represents the number of terms.
2. How do you calculate the sum of n terms of a geometric progression (GP)?
The sum of n terms of a geometric progression (GP), denoted as Sn, depends on the common ratio r.
• If r ≠ 1, the formula is: Sn = a(rn - 1) / (r - 1)
• If r = 1, the sum is simply na (since all terms are equal).
Where a is the first term and r is the common ratio.
3. What is the sum of the first n natural numbers?
The sum of the first n natural numbers is given by the formula: Sn = n(n + 1) / 2
4. Can the sum of n terms be negative?
Yes, the sum of n terms can be negative. This occurs when the terms of the sequence are predominantly negative, or when there's a combination of positive and negative terms resulting in a negative overall sum. For example, in an arithmetic progression with a negative common difference and a sufficiently large negative first term.
5. What does ‘n’ represent in the sum of n terms formulas?
In the formulas for the sum of n terms, 'n' represents the total number of terms being added in the arithmetic or geometric progression.
6. How is the sum of n terms formula derived for AP and GP?
The derivation involves using the general term formula and summing the series using either mathematical induction or the method of differences. A detailed explanation can be found in advanced textbooks and online resources, and is often part of a more detailed mathematical study of sequences and series.
7. When is it necessary to use the 'last term' formula for the sum of n terms?
The 'last term' formula, Sn = n/2(a + an), is useful when the last term (an) is known instead of the common difference (d). It simplifies calculation when the common difference is cumbersome to find.
8. How do infinite series sums differ from standard n term sums?
Standard n-term sums deal with a finite number of terms. Infinite series sums involve an infinite number of terms and can converge to a finite value or diverge to infinity (or negative infinity). Convergence depends on the specific series (e.g., geometric series converge if |r| < 1).
9. What are common calculation mistakes with these sum formulas in exams?
Common mistakes include:
• Incorrectly identifying the first term (a) or common difference (d) or common ratio (r).
• Substituting values into the wrong formula (AP vs. GP).
• Arithmetic errors in calculations involving exponents or fractions.
• Forgetting to check if the common ratio (r) is equal to 1 in a GP.
10. Can sum of n terms formulas be applied to non-numeric data?
While the formulas are primarily for numerical sequences, the underlying concepts can be adapted. For instance, if you have a sequence of categorical data with some sort of systematic progression, you might be able to model the 'sum' in a different way, but you would not use the arithmetic or geometric progression sum formulas directly.
11. What are the formulas for the sum of the squares and cubes of the first n natural numbers?
The sum of the squares of the first n natural numbers is: Σn² = n(n+1)(2n+1)/6
The sum of the cubes of the first n natural numbers is: Σn³ = [n(n+1)/2]²
12. How do I determine if a given sequence is an arithmetic progression (AP) or a geometric progression (GP)?
To identify whether a sequence is an arithmetic progression (AP) or a geometric progression (GP), examine the differences between consecutive terms:
• **AP**: If the difference between consecutive terms is constant, it's an AP. This difference is called the common difference (d).
• **GP**: If the ratio between consecutive terms is constant, it's a GP. This ratio is called the common ratio (r).
