Prove that sum of 1/2 n(n+1) and 1/2 (n+1)(n+2) is always a square

[lizhelpme]

Hi, I need to help my step son with his exams, I can answer most of the questions except this selection - can anyone help me?

1. n is an integer

prove algebraically that the sum of

1/2 n(n+1) and 1/2 (n+1)(n+2) is always a square number

2. show( 6-sqrt 8 )/(sqrt 2-1) can be written in form a + b sqrt 2

3. the ratio (y+x)y-x) is equivalent to k:1

show that y=x(k+1)/(k-1)

I will also attach the three questions with diagrams

THANK YOU MATHS GENIUS's!!!

 

[Deleted member 4993]

Hi, I need to help my step son with his exams, I can answer most of the questions except this selection - can anyone help me?

1. n is an integer

prove algebraically that the sum of

1/2 n(n+1) and 1/2 (n+1)(n+2) is always a square number

2. show( 6-sqrt 8 )/(sqrt 2-1) can be written in form a + b sqrt 2

3. the ratio (y+x)y-x) is equivalent to k:1

show that y=x(k+1)/(k-1)

I will also attach the three questions with diagrams

THANK YOU MATHS GENIUS's!!!

Please ask your step-son to contact us directly and tell us where he is exactly stuck.

 

[Dr.Peterson]

Hi, I need to help my step son with his exams, I can answer most of the questions except this selection - can anyone help me?

1. n is an integer

prove algebraically that the sum of

1/2 n(n+1) and 1/2 (n+1)(n+2) is always a square number

2. show( 6-sqrt 8 )/(sqrt 2-1) can be written in form a + b sqrt 2

3. the ratio (y+x) : (y-x) is equivalent to k:1

show that y=x(k+1)/(k-1)

1. I would just expand (multiply out, distribute) the expression 1/2 n(n+1) + 1/2 (n+1)(n+2), and then try to factor the result. It should end up factoring as a perfect square.

2. The standard thing to do here is to multiply numerator and denominator by the conjugate of the denominator, namely (sqrt(2) + 1).

3. You are given that (y+x) / (y-x) = k. You want to solve for y; a good first step is to multiply both sides by (y - x).

If you need more help than these starting hints, write back and show how far you can get.

 

[JeffM]

Alternatively, start factoring immediately

\(\displaystyle \dfrac{1}{2} * n(n + 1) + \dfrac{1}{2} * (n + 1)(n + 2) = \dfrac{1}{2} * (n + 1) * (n + n + 2) \implies \text {WHAT?}\)