Frankenodd Theory Of Odd Numbers

Dr. Frankenodd (brother of Frankenine) observed the following observations on numbers :

  • He took 3 consecutive positive integers and he added them. He noticed that all results were divisible by 3. \[\begin{align} {\color{blue}3 + 4 + 5} = {\color{red}12} = {\color{green}3 \times 4\\} \end{align}\]

  • He took 5 consecutive positive integers and he added them. He noticed that all the results were divisible by 5. \[\begin{align} {\color{red}43 + 44 + 45 + 46 + 47} = {\color{green}225} = {\color{blue}5 \times 45\\} \end{align}\]

  • He took 7 consecutive positive integers and he added them. He noticed that all the results were divisible by 7. \[\begin{align} {\color{green}22 + 23 + 24 + 25 + 26 + 27 + 28} = {\color{blue}175} = {\color{red}7 \times 25\\} \end{align}\]

Final Conclusion :

For \(n \text{ (such that n is a odd number)}\) consecutive positive integers the sum will always be a multiple of \(n\).

Will Dr. Frankenodd theory always works ?

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