Question: Which numbers can be or can not be expressed as a sum of two or more consecutive natural numbers. Let us divide the solution into a few small parts for quick and better understanding.
Part 1
First of all let us see which numbers can not be expressed as a sum of several consecutive natural numbers.
The answer lies in this fact: If we take two consecutive natural numbers; one will be odd and the other will be even, in either order.
But if we consider the numbers which are powers of 2 like 2, 4, 8,16, 32, 64 etc; they or their factors don't have any odd factor (other than 1) so they can not be expressed as a sum of two or more consecutive natural numbers.
So the necessary condition for a number to be expressed as a sum of several consecutive natural numbers is that it should have at least one odd factor, other than 1.
Part 2
Now let us come to odd numbers.
Every odd number n can be written as (2k + 1) which can further be written as k + (k+1). The numbers k and k+1 are two consecutive numbers.
So all odd numbers can definitely be expressed as a sum of two consecutive natural numbers. Thus all odd numbers are sorted.
Part 3
Now we are left with only the even numbers which have at least one odd factor other than 1. Or in other words, now let's find the even numbers that aren’t powers of two.
Clearly, such even numbers can only be the sum of a series of more than two consecutive numbers.
They must have an even number of odd numbers in the series of consecutive numbers, but any number of even numbers.
These even numbers are all of the form 2n.
If n (the half) is an odd number, n can be written as a pair of two consecutive numbers.
Then 2n can also be written as a sequence of four consecutive natural numbers, instead of two numbers.
To demonstrate this, let us take any even number that’s not the power of two and whose half is odd: Say 50.
It's half=25 which can be written as a sum of 12+13.
And 50 can be written as equal to 12+12+13+13 which can be rearranged as 11+12+13+14. We have balanced it by adding and subtracting 1 from two numbers at the end to get the desired result.
The same method of adding 1 and subtracting 1 several times can be applied to any such even number and some series of Consecutive Natural numbers can be easily found.
Part 4
In fact, we can generate a generalised formula to know the number of ways a number can be written as a sum of two or more consecutive natural numbers. Interestingly, we can also find ‘the ways’ too with a bit of mathematical jugglery.
I have already given the formula way back in 2013 to my beloved students who were also in pursuit of ‘Joy of Mathematics’ apart from being aspirants for CAT-IIM.
It will require revisiting Number Theory and Factor theory. You may wish to click here to revisit my blog post ( Short Cuts for Numbers) on Number theory to understand the formula. The formula is given in the hand written note at the serial number 17.
We can discuss its full derivation in a separate blog post once you become familiar with the first sixteen formulae as given in the above mentioned blog post.
(You may also wish to solve a few questions on numbers from the Blog Post: Investigations of a Number 420.)
If sufficient friends gather, I can pick up chalk once again for a day or two in a week at any place in Indore.
Let the conventional Joy of Mathematics prevail forever…
