Sum of First n Natural Numbers Using the n(n+1)/2 Formula

Sum of First n Natural Numbers means adding the integers from 1 to n. The standard formula is n(n + 1)/2. This page explains its derivation, direct calculation method, pairing shortcut, related sums, and solved examples useful for quantitative aptitude questions and number-series problems.

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What Is the Sum of the First n Natural Numbers?

The sum of the first n natural numbers is the total of 1 + 2 + 3 + ... + n, and it equals n(n + 1)/2.

Here, n is the last natural number included in the sum. For example, the sum of the first 10 natural numbers is 10 × 11 ÷ 2 = 55. The sequence has n terms, with first term 1 and last term n.

Sum of First n Natural Numbers Video Lessons

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Special Sums: n, n² and n³ Formulas

Learn the formulas for the sums of the first n natural numbers, their squares, and their cubes, with a clear understanding of how these special summations are represented.

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Sum of First n Natural Numbers Formula & Tricks

Important Formulas

Standard formula
1 + 2 + 3 + ... + n = n(n + 1)/2

Use this formula when the sequence starts at 1 and ends at n.

Arithmetic-series form
Sₙ = n/2 × (first term + last term) = n(n + 1)/2

For the sequence 1, 2, ..., n, the first term is 1 and the last term is n.

Number of terms in a consecutive range
Number of terms from a to b = b − a + 1

This formula helps find the term count before applying the arithmetic-series sum formula.

Sum from a to b
a + (a + 1) + ... + b = (b − a + 1)(a + b)/2

Use this general form for consecutive integers that do not begin with 1.

Quick Tricks

Pair the first and last terms

In 1 + 2 + ... + n, pair terms from the two ends. Every pair has sum n + 1. If n is even, there are n/2 pairs. If n is odd, pair all but the middle term and add the middle term separately.

Example: For 1 to 10, five pairs each total 11, so the sum is 5 × 11 = 55.
Use consecutive-sum blocks

To find the sum from a to b, count the terms and multiply the term count by the average of the first and last terms.

Example: The sum from 21 to 30 has 10 terms and average (21 + 30)/2 = 25.5. Therefore, the sum is 10 × 25.5 = 255.
Use the triangular-number pattern

The sums of the first natural numbers are triangular numbers. The next sum is found by adding the next integer to the previous sum.

Example: If the sum from 1 to 8 is 36, then the sum from 1 to 9 is 36 + 9 = 45.

Sum of First n Natural Numbers Concepts

Derivation by Pairing

The formula n(n + 1)/2 follows by adding the sequence forward and backward.

Let S = 1 + 2 + 3 + ... + n. Writing it in reverse gives S = n + (n − 1) + ... + 1. Adding the two equations produces 2S = n(n + 1), because there are n pairs and each pair equals n + 1. Hence, S = n(n + 1)/2.

Example: For n = 6, 2S = 6 × 7 = 42, so S = 21.

Using the Arithmetic-Series Formula

The sum of consecutive integers equals the number of terms multiplied by their average.

For an arithmetic progression, S = number of terms × (first term + last term)/2. In 1, 2, ..., n, the number of terms is n, the first term is 1, and the last term is n. Therefore, S = n(1 + n)/2.

Example: For 1 to 15, S = 15 × (1 + 15)/2 = 15 × 8 = 120.

Sum of a Consecutive Range

The sum from a to b is (b − a + 1)(a + b)/2.

First find the number of terms, b − a + 1. Then find the average, (a + b)/2. Multiplying these quantities gives the required sum. This method works when a and b are integers and every integer between them is included.

Example: The sum from 12 to 18 is (18 − 12 + 1)(12 + 18)/2 = 7 × 15 = 105.

Finding n from a Given Sum

If the sum of the first n natural numbers is S, solve n(n + 1)/2 = S.

The equation becomes n² + n − 2S = 0. Select the positive integer solution because n represents the number of natural numbers. For a valid triangular number, this solution is a positive integer.

Example: If S = 78, then n(n + 1)/2 = 78, so n(n + 1) = 156. Since 12 × 13 = 156, n = 12.

Sum of First n Natural Numbers Data Table

Use the structured data below while following the explanation and examples.

Common sums of the first n natural numbers

Values obtained using S = n(n + 1)/2.

n Expression Sum
5 1 + 2 + 3 + 4 + 5 15
10 1 + 2 + ... + 10 55
15 1 + 2 + ... + 15 120
20 1 + 2 + ... + 20 210

Note: The value of n is the last term and also the number of terms.

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Practice Sum of First n Natural Numbers Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Sum of First n Natural Numbers: Revision Points

Use these formulas and rules for quick revision.

  • The first n natural numbers are 1, 2, 3, ..., n.
  • Their sum is n(n + 1)/2.
  • For a range from a to b, use (b − a + 1)(a + b)/2.
  • The number of integers from a to b is b − a + 1.
  • Pairing the first and last terms gives the constant pair sum n + 1.
  • To find n from a sum S, solve n(n + 1)/2 = S and retain the positive integer solution.
  • The sum of the first n natural numbers is also called the nth triangular number.

Sum of First n Natural Numbers FAQs

What is the sum of the first 25 natural numbers?

Use n(n + 1)/2: 25 × 26 ÷ 2 = 325.

What is the sum of the first 100 natural numbers?

The sum is 100 × 101 ÷ 2 = 5,050.

How do you find the sum from 10 to 25?

There are 25 − 10 + 1 = 16 terms. The sum is 16 × (10 + 25)/2 = 280.

What is the sum of the first 50 odd natural numbers?

The sum of the first n odd natural numbers is n². Therefore, the sum of the first 50 odd numbers is 50² = 2,500.

What is the sum of the first 20 even natural numbers?

The first 20 even numbers are 2, 4, ..., 40. Their sum is 20(20 + 1) = 420.

How can you determine n when the sum is 210?

Set n(n + 1)/2 = 210, giving n(n + 1) = 420. Since 20 × 21 = 420, n = 20.

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