Sum of consecutive odd numbers

 
Parity
http://mathworld.wolfram.com/Parity.html

 

Let   n   be a positive integer and   m   be any integer with the same parity as   n
with   m \; \geq  \; n

These odd integers are all positive.

 

for example,

If   n = 2,    m = 4

m \times n = 8

8   =   3 + 5

the product is equal to the sum of 2   (n = 2)   consecutive odd integers

 

If   n = 3,   m = 5

m \times n = 15

15   =   3 + 5 + 7

the product is equal to the sum of 3   (n = 3)   consecutive odd integers

 

If   n = 5,    m = 7

m \times n = 35

35   =   3 + 5 + 7 + 9 + 11

the product is equal to the sum of 5   (n = 5)   consecutive odd integers

 

Can you prove the product   m \times n   will always equal to the sum of   n   consecutive odd integers?

 

                                                             ——————————————
 

Proof   –  

 

The sum of the first   n   odd integers is   n^2 :

1 \; + \; 3 \; + \; 5 \; + \; ... \; + \; (2 n-1) \; = \; n^2

From this, we have that the sum of any sequence of consecutive odd integers is :

(2m+1) + (2m+3) + ... + (2n-1)

= \; (1 + 3 + 5 + ... + (2n-1)) \; - \; (1 + 3 + 5 + ... + (2m-1))

= \; n^2 \; - \; m^2

n^2 \; - \; m^2 \; = \; (n+m) (n-m)

So an integer   k   can be written as the sum of consecutive odd integers if it can be expressed as

k \; = \; a b

where

a \; = \; n+m        and        b \; = \; n-m

a+b \; = \; 2 n        or        n \; = \; (a+b)/2

a-b \; = \; 2 m        or        m \; = \; (a-b)/2

 

a   is odd,    b   is odd :

If   k \; = \; 3 \times 5 \; = \; 15

n \; = \; (5+3)/2 = 4        m \; = \; (5-3)/2 = 1

The first odd integer in the sequence is:   2(1) + 1 = 3

The last odd integer in the sequence is   2(4) – 1 = 7

We have the solution   15 \; = \; 3+5+7

 
a   is even,    b   is even :

E.g.   12 \times 6 \; = \; 72

a = 12,    b = 6

n = (a+b)/2         n = (12+6)/2 = 9
m = (a-b)/2         m = (12-6)/2 = 3

1st odd integer is :      2(3+1) – 1 = 7
last odd integer is :      2(9) – 1 = 17

We have the solution   7 + 9 + 11 + 13 + 15 + 17   =   72

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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