The difference of the differences of the squares of any two consequtive numbers is ?

It's always 2....... And alsooo the consecutive differences of squares of two consecutive numbers are always consecutive odd numbers.....

let the first number be defined as x and the second one as y, where x and y are whole numbers and y=x+1.

then, Y^2 - X^2 = (Y - X)(Y + X)

since Y=X+1 => (X + 1 - X)(Y + X)

=> (1)(Y+X)

this is the mathematical proof that the difference of the squares of two whole numbers is equal to the sum of the numbers

as for the other answers you have to be careful and do not make mistakes when working with negative number ( the 2a+1 example )

enable x be the extra beneficial form and y the smaller. Then: x - y = 3 x^2 - y^2 = 39 y = x - 3 So x^2 - (x - 3)^2 = 39 x^2 - x^2 + 6x - 9 = 39 6x = 40 8 x = 8 to verify: y = 5 8^2 = sixty 4, 5^2 = 25 and sixty 4 - 25 = 39 as required = A

It's "2" the difference of the differences of the squares of two consecutive numbers.

eg: 2^2-1^2=3

3^2-2^2=5

5-3=2

The difference of the the squares of any two consecutive numbers is the sum of the two mumbers.

But what you have asked is something else which I don't understand. Maybe you meant the 'difference of the differences' of the squares of three consecutive numbers. In this case, the answer is two.

This question is in the form of b^2-a^2 where b=a+1.

b^2-a^2=

(a+1)^2-a^2

a^2+2a+1-a^2

2a+1

so that answer is that you take the smaller number double it and add one.

Exmaple Let a=5 and b=6 accourding to the formula the answer should be 2(5)+1=11

b^2-a^2=

6^2-5^2

36-25

11

the sum of the 2 numbers

if n is the smaller number,

the next consecutive number is n+1

so square of (n+1) = nsquare+2n+1

square of n = nsquare

so difference = 2n+1

so the answer for

difference of the differencesof any two consecutive numbers is

2 times the smaller number plus one

choose two consecutive numbers

a+b and a+b+1

squares are a^2+2ab+b^2 and a^2+b^2+ 2ab +2a +2b +1

difference between squares is 2a+2b+1

now choose two different consecutive numbers

a+c and a+c+1

the difference between the difference of the squares is

|2b-2c| independent of a.

now set b=0 ...

the difference will be the sum of the two numbers.