Algebraic argument & proof

Prove that the difference in the squares of two consecutive even numbers is always a multiple of 4

There are only red counters and yellow counters in bag A. number of red counters: number of yellow counters = 3: 5 There are only green counters and blue counters in bag B. The number of counters in bag B is half the number of counters in bag A. Given that there are x red counters in bag A, use algebra to show that the total number of counters in bag A and bag B is 4x

13An expression for the nth term of the sequence of triangular numbers is Prove that the sum of any two consecutive triangular numbers is a square number.

Factorise a² – b²

Show that 2⁴⁰ – 1 is the product of two consecutive odd numbers.

Prove that (2m + 1)² − (2n – 1)² = 4(m + n)(m − n + 1)

Sophia says that the result in part (a) shows that the difference of the squares of any two odd numbers must be a multiple of 4

Is Sophia correct? You must give reasons for your answer.

A right-angled triangle is formed by the diameters of three semicircular regions, A, B and C as shown in the diagram.

Show that area of region A = area of region B + area of region C

Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4

Given that n can be any integer such that n 1, prove that n² – n is never an odd number.

Prove algebraically that the difference between the squares of any two consecutive odd numbers is always a multiple of 8

Prove that the square of an odd number is always 1 more than a multiple of 4

n is an integer.

a and b are odd numbers.

Give an example to show that the value of 2(a + b) is a multiple of 4

Show that, when a and b are both odd numbers, the value of 2(a + b) will always be a multiple of 4