Question

N is an integer

prove algebraically that the sum of 1/2n(n+1) and 1/2(n+1)(n+2) is always a square number

Answer 1

Work:

• [1/2(n)(n + 1)] + [1/2(n + 1)(n + 2)]

Simplifying in Brackets

• => [1/2(n² + n)] + [1/2(n² + 2n + n + 2)]
• => [n²/2 + n/2] + [n²/2 + n + n/2 + 1]

Removing Brackets.

• => n²/2 + n/2 + n²/2 + n + n/2 + 1

Solving.

• => n² + n + 1
• => (n + 1)²

Since the result has a "square" sign, the statement is proved has true.

Hoped this helped.

Answer 2

• See below

Explanation:

Find the sum:

• 1/2n(n+1) + 1/2(n+1)(n+2) =
• 1/2(n² + n) + 1/2(n² + 3n + 2) =
• 1/2(n² + n + n² + 3n + 2) =
• 1/2(2n² + 4n + 2) =
• n² + 2n + 1 =
• (n + 1)²

Proved