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Question 1: Prove that the difference of the squares of 2 consecutive numbers is always the sum of the 2 numbers |

Question 2: Prove that (5n+8)^2 +(5n-8)^2 is always even |

Question 3: Prove that the product of 2 consecutive numbers is always even |

Question 4: Prove the identity (n+5)^2 -(6n+85) ≡ (n+10)(n-6) |

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Question 1: (2n+1)^2-(2n)^2 =4n^2+4n+1-4n^2 =4n+1=(2n+1)+(2n) therefore the difference of the squares of 2 consecutive numbers is always the sum of the 2 numbers |

Question 2: (5n+8)^2 +(5n-8)^2 =(25n^2+80n +64)+(25n^2-80n +64) =50n^2+128=2(25n^2+64) (5n+8)^2 +(5n-8)^2 is a multiple of 2, therefore it is even |

Question 3: 2n(2n+1) =2(2n^2+n) Any multiple of 2 is always even, therefore the product of 2 consecutive numbers is always even |

Question 4: (n+5)^2 -(6n+85) =(n^2+10n +25)-(6n+85) =n^2+4n-60 =(n+10)(n-6) |