Algebraic argument & proof

Prove that, for any three numbers which are consecutive multiples of 4, the difference between the square of the largest number and the square of the smallest number is always a multiple of 64

Show clear algebraic working.

Prove algebraically that, for any three consecutive even numbers,

the sum of the squares of the smallest even number and the largest even number is 8 more than twice the square of the middle even number.

The diagram shows a solid cone and a solid sphere.

The cone has base radius r, slant height l and perpendicular height h The sphere has radius r

The base radius of the cone is equal to the radius of the sphere.

Given that

k volume of the cone = volume of the sphere

show that the total surface area of the cone can be written in the form

where a is a constant to be found.

Prove that when the sum of the squares of any two consecutive odd numbers is divided by 8, the remainder is always 2 Show clear algebraic working.

Prove algebraically that the product of any two odd numbers is always an odd number.

N is a multiple of 5

A = N + 1 B = N – 1

Prove, using algebra, that A² – B² is always a multiple of 20

The table gives information about the first six terms of a sequence of numbers.

Term number	1	2	3	4	5	6
Term of sequence	1 2 2	2 3 2	3 4 2	4 5 2	5 6 2	6 7 2

Prove algebraically that the sum of any two consecutive terms of this sequence is always a square number.

Show that x(xx + 1) = x³x

Prove that the difference between a whole number and the cube of this number is always a multiple of 6