We will activate by demography a attending at odd numbers. We all apperceive what they are: 1, 3, 5, 7, 9, and so on.

They are the numbers that don’t brace up evenly. If we bisect an odd cardinal by 2, we will accept one larboard over.

Odd or even?

Pair them up and it is accessible to tell…

This is an odd cardinal of squares.

Mathematically, we can accomplish an odd cardinal by abacus some accumulation by two, and either abacus or abacus one. Multiplying by two gives us commutual up numbers, and abacus or abacus one makes the cardinal odd.

So the analogue of an odd cardinal could be

Odd cardinal = 2N 1 or Odd cardinal = 2N -1

If we bind N to the counting numbers (1, 2, 3, …) so we accept a calculation of the cardinal of Odd numbers, afresh we will accept to use Odd cardinal = 2N – 1, otherwise, our aboriginal odd cardinal would be three instead of one. Computer programmers, who alpha counting from zero, can use 2N 1, but for the blow of us mortals, adage the zeroth odd cardinal is one aloof doesn’t complete right.

Here are the aboriginal four odd numbers, 1, 3, 5 and 7

We will add them up, and see what we get. N is the cardinal of Odd numbers we add up.

N= 1: 1 = 1 N= 2: 1 3 = 4 N= 3: 1 3 5 = 9 N= 4: 1 3 5 7 = 16

There seems to be a pattern. Let’s booty a attending at it application diagrams

N=1:

N=2:

N=3:

N=4

The arrangement is the sum of all the odd numbers (starting at one) is N squared.

The appropriate attractive attribute on the larboard of the according assurance is an abridgement for accretion up a account of numbers. The “1” beneath the attribute (Capital Sigma, in Greek, by the way) gives the starting number, and the “N” on top gives the catastrophe number. After the assurance you acquisition a arrangement or announcement for addition out what to add up, in this case, odd numbers.

So, what we are absolutely adage is that

Can we prove it?

The aboriginal footfall in the affidavit is already done. We can attending at the archetype and see that the blueprint works for N=1, N=2, N=3, N=4.

The additional footfall is to appearance that we can go from some approximate N to N 1. If we can do this, we accept apparent that we can alpha on any of our aboriginal four steps, and go as aerial as we please. To infinity, as Buzz Lightyear says, and beyond.

We activate by afterlight Nth Odd Cardinal application our beforehand analogue for Odd Numbers:

Nth Odd cardinal = 2N – 1

Now, we accept that the blueprint is true, so we can say

and we appetite to acquisition out whether abacus the aing odd number, 2(N 1)-1 agency the sum is the aboveboard of N 1

We can alter the alpha series,

So

We will use the distributive acreage on the larboard duke ancillary to aish the departure about N 1, and carbon the appropriate duke ancillary after the exponent.

Next, we can do the bit of addition on the larboard side, 2-1, and use the distributive acreage to aish one set of parentheses on the appropriate side.

Using the distributive acreage afresh on the appropriate ancillary gives

And we can add up the N N to get 2N on the appropriate side, which after-effects in

Once we see the being on the larboard ancillary is the aforementioned as the being on the appropriate side, we can say the blueprint is true. Which agency the blueprint we started with is true, too.

So this shows we can get from any footfall to the aing college footfall in our alternation of sums of odd numbers. Since we apperceive that the blueprint works for the aboriginal four steps, we accept accepted it.

Have fun in the comments.

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