Pythagorean triples and number patterns

A question is this type if and only if it involves proving properties of Pythagorean triples or showing that certain integer expressions form specific patterns (e.g., 2t, t²-1, t²+1).

2 questions · Standard +0.3

1.01a Proof: structure of mathematical proof and logical steps
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OCR MEI C3 2006 June Q5
6 marks Standard +0.3
5 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
  1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).
OCR MEI C3 Q13
6 marks Standard +0.3
Positive integers \(a\), \(b\) and \(c\) are said to form a Pythagorean triple if \(a^2 + b^2 = c^2\).
  1. Given that \(t\) is an integer greater than 1, show that \(2t\), \(t^2 - 1\) and \(t^2 + 1\) form a Pythagorean triple. [3]
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2t\), \(t^2 - 1\) and \(t^2 + 1\). [3]