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The sum of the first n odd numbers is always a square

I was staring at the checked pattern on the back of an airline seat the other day when I suddenly saw that the sum of the first n odd numbers is always a square. For example,

1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16

And, of course, it occurred to me that it would be nice to be able to prove it. There are lots of ways to do that. Firstly, this is just the sum of an arithmetic progression starting at a = 1 with a difference of d = 2. So the standard formula gives us:

sum_odd(n) = n(2a + (n-1)d)/2
= n(2 + (n-1)2)/2
= n(1 + n - 1)
= n^2

So, the sum of the first n odd numbers is n^2.

But using standard formulae is annoying, so how about trying a little induction.

sum_odd(1) = 1

sum_odd(n+1) = sum_odd(n) + (2n + 1)
= n^2 + 2n + 1
= (n+1)^2

But back to the airline seat. Here's what I saw (I added the numbering, Lufthansa isn't kind enough to do that for you :-):



The other thing I noticed was this:



You can view the square as the sum of two simpler progressions (the sum of the first n numbers and the sum of the first n-1 numbers):

1 + 3 + 5 + 7 =
1 + 2 + 3 + 4 +
1 + 2 + 3

And given that we know from Gauss the sum of the first n numbers if n(n+1)/2 we can easily calculate:

sum_odd(n) = sum(n) + sum(n-1)
= n(n+1)/2 + (n-1)n/2
= (n^2 + n + n^2 - n)/2
= n^2

What do you do on long flights?

Comments

Unknown said…
Just noticing that they agree on the first three values

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9

is actually a valid proof. $\sum_{i=1}^n p(i)$, where $p(i)$ is a polynomial of degree $d$, is a polynomial of degree at most $d + 1$ (you can prove this by induction). Now $n^2$ and $\sum_{i=1}^n (2n -1)$ are both polynomials of degree at most 2, and they agree on three values, hence must be identical.
James said…
What do you do on long flights?
Watch movies...
turd said…
The sum of the first n odd numbers is = to n^2. Im in 7th grade and I know that
turd said…
the sum of the first n odd numbers is n^2. U ok sir im in 7th grade and i know that
turd said…
The sum of the first n odd numbers is = to n^2. Im in 7th grade and I know that
Unknown said…
Find the sum of first n odd numbers............1,3,5,7,.........2n-1........will you please give me the answer????? I am 10 std student
Unknown said…
Write a C program to take the lower limit and upper limit of the range as inputs and calculate!

i. Square sum of those interval numbers
ii. Show the odd numbers only within the interval
iii. Determine the average of those numbers.
Unknown said…
This comment has been removed by the author.
Unknown said…
Just to make it more fun
1=1^3
3+5=2^3
7+9+11=3^3

Proof isn't quite so pretty, but you can notice their average is always n^2
Unknown said…
Please answer: the sum of odd numbers from 10 to 60 is??
Unknown said…
25/2*(11+59)
Since 25 odds from 10 to 60
=875

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