# Prove that the sum of 6 consecutive odd numbers is an even number?

##### 4 Answers

Please see below.

#### Explanation:

Any two consecutive odd numbers add up to an even number.

Any number of even numbers when added result in an even number.

We can divide six consecutive odd numbers in three pairs of consecutive odd numbers.

The three pair of consecutive odd numbers add up to three even numbers.

The three even numbers add up to an even number.

Hence, six consecutive odd numbers add up to an even number.

Let first odd number be

Six consecutive odd numbers are

#(2n-1),(2n+1),(2n+3), (2n+5),(2n+7),(2n+9)#

Sum of these six consecutive odd numbers is

#sum=(2n-1)+(2n+1)+(2n+3)+ (2n+5)+(2n+7)+(2n+9)#

Adding by brute force method

#sum=(6xx2n)-1+1+3+5+7+9#

We see that first term will always be even

#=>sum="even number"+24#

Since

#:.sum="even number"#

Hence Proved.

See below

#### Explanation:

An odd number has the form

Let be the first

We know also that the sum of n consecutive numbers in a arithmetic progresion is

which is an even number for every

#### Explanation: