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I am doing some questions on:

  1. Obtaining the formula for the sum of the first $n$ odd natural numbers which I have got as $n^2$

  2. Obtaining the formula for the sum of the first $n$ even natural numbers which I have got as $n(n+1)$

I am now supposed to use my answers to complete the third question, with which I am struggling:

  1. What is the sum of the first $m$ natural numbers, both even and odd?

Any advice on how to go about this question?

aelcro
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    Hint: consider cases $n = 2k$ and $n = 2k+1$ - how many odd and even numbers will there be in both cases? What will be there sum? – mihaild Nov 09 '22 at 16:22
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    How did you solve problems 1 and 2? Why didn't that work for problem 3? – Arthur Nov 09 '22 at 16:22
  • @JMoravitz I don't think it does, because OP (strangely) wants to deduce 3 from 1 and 2, instead of finding it directly. But mihaild's comment above gave the appropriate clue. – Anne Bauval Nov 09 '22 at 16:38
  • The easiest method is probably the pairing method. One writes the sum $$S=1+2+3+\cdots +(n-2)+(n-1)+n$$ backwards $$S=n+(n-1)+(n-2)+\cdots +3+2+1$$ and notices easily $$2S=(n+1)\cdot n$$ since there are $n$ pairs summing up to $n+1$. – Peter Nov 09 '22 at 17:11
  • @mihaild Thank you! I've got it now. – aelcro Nov 10 '22 at 14:26

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