Consecutive natural numbers
An investigation into natural numbers
1. Write a definition of the term ‘consecutive natural numbers’. (Don’t include the word
consecutive in your definition and if you use the word integer then write what that
means too!)
2. Every odd number can be written as the sum of two consecutive numbers.
e.g.
3=1+2
5=2+3
Write at least five examples to demonstrate this.
3. Design a method to show how to write every odd number as the sum of two consecutive
numbers.
4. A positive integer can be written as a sum of consecutive natural numbers in as many
different ways as it has odd factors greater than 1.
Each of the following numbers below has three odd factors greater than 1. Find the
three different ways each can be written as sum of consecutive numbers.
a. 15
b. 21
c. 27
5. Look at these natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20…
Some of them can’t be written as the sum of consecutive integers. Which ones?
6. The sum of the first twelve positive integers can be written as the sum of which two
consecutive prime numbers?
7. Find a quick way to sum 1 + 2 + 3 +.............+ 1000
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Consecutive natural numbers
Teacher notes
This activity would work best in either pairs or small groups (perhaps Kagan pairs, then moving
to Kagan fours.)
Students must understand the definitions of the key words (consecutive, natural number,
integer, positive, sum, prime) before moving on the other activities so this could possibly be a
starter. (Think, pair, share?)
Answers
1. A natural number is a positive integer.
2. Many solutions
3. Pick any odd number: 35
Halve it: 17.5
Subtract 0.5 and add 0.5 to give the two solutions: 17 + 18
4. a.
b.
c.
15 = 7 + 8 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5
21 = 10 + 11 = 6 + 7 + 8 = 1 + 2 + 3 + 4 + 5 + 6
27 = 13 + 14 = 8 + 9 + 10 = 2 + 3 + 4 + 5 + 6 + 7
5. The powers of 2 can never be written as the sum of consecutive numbers because they
have no odd factors.
6. 1 + 2 +......+ 11 + 12 = 78 = 37 + 41
7. 1 + 2 + 3 +.............+ 998 + 999 + 1000
Each coloured pair totals 1001 and there are 500 pairs.
The sum is 1001 x 500 = 500500
From an anecdote about Gauss:
Karl Guass’ teacher wanted some time to himself so he set his pupils the task of summing the
first 100 natural numbers. To the teacher’s annoyance, Karl produced the answer in seconds,
using the method above.
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