# Unit Fraction Uniqueness 2

## When you add unit fractions with consecutive denominators, something quite interesting happens! Can you figure out the pattern to the table below? The last row will show you the formula!

Fraction #1 | Fraction #2 | Sum |

1/2 | 1/3 | 5/6 |

1/3 | 1/4 | 7/12 |

1/4 | 1/5 | 9/20 |

1/5 | 1/6 | 11/30 |

1/6 | 1/7 | 13/42 |

1/7 | 1/8 | 15/56 |

1/8 | 1/9 | 17/72 |

1/9 | 1/10 | 19/90 |

1/x | 1/(x+1) |
(2x+1)/(x^2+x) |

## You get the sum of the denominators as the numerator & the product as the denominator in the sum!

### Since addition is commutative, swapping Fractions #1 & #2 will of course give you the same thing!

Fraction #1 | Fraction #2 | Sum |

1/3 | 1/2 | 5/6 |

1/4 | 1/3 | 7/12 |

1/5 | 1/4 | 9/20 |

1/6 | 1/5 | 11/30 |

1/7 | 1/6 | 13/42 |

1/8 | 1/7 | 15/56 |

1/9 | 1/8 | 17/72 |

1/10 | 1/9 | 19/90 |

1/(x+1) | 1/x | (2x+1)/(x^2+x) |

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© Derek Cumberbatch