A mathematical evidence is a sequence of statements the follow on logically from each various other that mirrors that miscellaneous is 6294.orgnstantly true. Utilizing letters to stand for numbers way that we can make statements about all numbers in general, quite than details numbers in particular.

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## Odd and even numbers

If (n) is an creature (a whole number), then the expression (2n) to represent an also number, since even numbers space the multiples the 2. The expression (2n - 1) and also (2n + 1) deserve to represent strange numbers, as an strange number is one less, or one more than an even number.

### Example

Prove the whenever two even numbers room added, the full is additionally an even number.

Try some examples: (2 + 2 = 4), (4 + 12 = 16), (1002 + 3024 = 4026).

This reflects that the declare is true because that these examples, but to prove the it is true all the time we have the right to use algebra.

Write the an initial of the two also numbers together (2n), where (n) is an integer. The 2nd even number have the right to be 6294.orgmposed as (2m), wherein (m) is additionally an integer. Us can’t use the very same letter for both expressions, otherwise castle would stand for the same even number.

Adding the two also numbers offers (2n + 2m).

This have the right to be factorised to offer (2n + 2m = 2(n + m))

Since (n) and (m) space both integers, then (n + m) will additionally be one integer, so the expression (2(n + m)) to represent an even number.

This reflects that whenever two even numbers space added, the 6294.orgmplete is additionally an also number due to the fact that (2n + 2m = 2(n + m)).

### Example

Prove that the product of 2 odd number is always odd.

Product is the value obtained by multiplying. Shot some examples: (3 imes 3 = 9), (7 imes 9 = 63), (11 imes 13 = 143).

For these examples, strange x odd = odd. To prove the it is true for every odd numbers, we have the right to write 2 odd numbers together (2n + 1) and also (2m + 1), where (n) and (m) are integers.

Multiplying the 2 odd numbers with each other gives:

<(2n + 1)(2m + 1) = 4nm + 2n + 2m + 1>

The an initial three terms have a typical factor that 2, so the expression deserve to be re-written as:

<4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1>

Since (n) and also (m) are integers, the expression inside the bracket, (2nm + n + m), will additionally be an integer. This way that the expression (2(2nm + n + m) + 1 ) to represent an odd number, together it is 2 multiply by an integer add to 1.

So the product of two odd numbers is always odd because ((2n + 1)(2m + 1) = 2(2nm + n + m) + 1).

Question

(a) is one odd number. Prove that (3a + 2) is always an weird number.

Since (a) is an odd number, we deserve to write (a = 2n + 1), wherein (n) is one integer.

Substituting this expression and simplifying gives

<3a + 2 = 3(2n + 1) + 2>

<= 6n + 3 + 2>

<= 6n + 5>

We require to show that this is one odd number, which method we room able 6294.orgme rearrange the expression right into the kind (2m + 1), whereby m is an integer. To acquire the “+1” term, we deserve to re-write the expression as:

<6n + 5 = 6n + 4 + 1>

The an initial two regards to this expression, (6n + 4) have a usual factor that 2, therefore factorising gives:

<6n + 5 = 6n + 4 + 1 = 2(3n + 2) +1>

Since (n) is one integer, the term in the bracket, (3n + 2) will additionally be an integer, so us have displayed that (3a + 2) is 6294.orgnstantly an odd number, as it have the right to be 6294.orgmposed in the form (2m + 1), wherein (m) is one integer.

## 6294.orgnsecutive integers

6294.orgnsecutive integers are entirety numbers the follow each various other without gaps. For example, 15, 16, 17 room 6294.orgnsecutive integers. If (n) is one integer, climate the 6294.orgnsecutive integers starting at (n) are (n), (n + 1), (n + 2), (n + 3) and so on.

### Example

Prove that the sum of 3 6294.orgnsecutive integers is a many of 3.

Try part examples: (1 + 2 + 3 = 6), (5 + 6 + 7 = 18), (102 + 103 + 104 = 309). This reflects the sum of three 6294.orgnsecutive integers is a lot of of 3 in these cases, yet to prove the is true in every cases, we can use algebra.

We can write 3 6294.orgnsecutive integers together (n), (n + 1) and (n + 2), therefore the sum of 3 6294.orgnsecutive integers can be 6294.orgmposed as: (n + (n + 1) + (n + 2))

Simplifying this expression gives:

This have the right to be factorised to provide (3n + 3 = 3(n + 1)) which will certainly be a lot of of 3 for every integer worths of (n).

Question

Prove the the difference in between two 6294.orgnsecutive square number is always an weird number.