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Lesson 5 of 8

Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 5 of 8

Sum of the First n Natural Numbers

“Adding 1 through n looks exhausting until a clever formula does the heavy lifting.”

Learning Objectives

• Derive the sum of the first n natural numbers. • Use the pairing method. • Understand the visual rectangular-array argument. • Apply the formula to consecutive-number sums. • Connect the formula with triangular numbers.

First n Natural Numbers

Adding consecutive numbers such as 1 + 2 + 3 + 4 + … one term at a time is easy when there are only a few terms. But as the number of terms becomes larger, this method quickly becomes slow and inconvenient. Finding the sum of the first 100 or 1000 natural numbers by direct addition would take a lot of unnecessary work.

A much better approach is to look for a pattern in the way the numbers can be grouped. By pairing terms from the beginning and the end of the sequence, we can create equal sums and use them to build a simple formula. This formula allows us to find 1 + 2 + 3 + … + n directly for any positive integer n, without adding every term separately.

The Pairing Idea

Let S=1+2+3+…+n. Write the same sum backwards underneath it: S=n+(n−1)+(n−2)+…+1. Every vertical pair now adds to n+1, and there are n such pairs.

Doubling the sumLaTeX
Sum of first n natural numbersLaTeX
First 100 Natural Numbers

Problem
Find 1+2+3+…+100.

  1. 1.Use Sₙ=n(n+1)/2.
  2. 2.S₁₀₀=100×101/2.
  3. 3.S₁₀₀=5050.
Two triangular arrays forming a rectangle Two triangular arrays of twenty-one dots are combined to form a rectangle containing six rows and seven columns, illustrating that two times S equals six times seven. S = 1 + 2 + 3 + 4 + 5 + 6 7 columns 6 rows First triangular array: S Second triangular array: S 2S = 6 × 7 = 42, therefore S = 21
Two triangular arrays forming a rectangle

Consecutive Numbers Not Starting from 1

To find a sum such as 25+26+…+58, subtract two natural-number sums: sum to 58 minus sum to 24.

Consecutive Sum

Problem
Find 31+32+…+60.

  1. 1.Sum 1 to 60 =60×61/2=1830.
  2. 2.Sum 1 to 30 =30×31/2=465.
  3. 3.Required sum=1830−465=1365.

Triangular Numbers Reappear

The nth triangular number is exactly the sum 1+2+…+n, so triangular numbers have the explicit formula n(n+1)/2.

nth triangular numberLaTeX

Practice Problems

Practice Problems
  1. Find the sum of the first 25 natural numbers.
  2. Find 1+2+…+75.
  3. Find 18+19+…+42 using subtraction of two natural-number sums.
  4. Find the 20th triangular number.
  5. A child arranges marbles in 30 rows, with 1 marble in the first row, 2 in the second, and so on. How many marbles are used?
  6. Find the smallest n for which 1+2+…+n is greater than 500.

Key Takeaways

Key Takeaways

• Pairing the sum forwards and backwards creates equal pairs. • Sₙ=n(n+1)/2. • Consecutive-number sums can be found by subtraction. • Triangular numbers use the same formula. • A visual two-triangle arrangement explains why the formula works.

Coming Next

Next, we study geometric progressions, where terms grow or shrink by multiplying by the same factor.

Previous · Lesson 4

Arithmetic Progressions

Next · Lesson 6

Geometric Progressions