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Proof: Proceed by induction

Base case $P(t=1)=n_1$, which is even since $n$ is an even integer. True. Assume $P(t)$, i.e. if $n_1,n_2,\cdots n_t$ are even, then $n_1+n_2+...+n_t$ is even, is true.

Inductive step: $P(t+1)=n_1+n_2+...n_t+n_{t+1}$. Since $n_1+n_2+...+n_t$ is assumed to be even and $n_{t+1}$ is even, an even $+$ even gives an even number. Therefore $P(t+1)$ is shown to be true by induction.

Please let me know if this proof is satisfactory as I am relatively new to induction proofs.

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    $\begingroup$ Satisfactory, but your MathJax isn't. $\endgroup$ – Parcly Taxel Feb 28 '18 at 3:59
  • $\begingroup$ For reference: Mathjax tutorial $\endgroup$ – JMoravitz Feb 28 '18 at 4:10
  • $\begingroup$ Don't you need to prove that the sum of $t$ terms that are even is itself even rather than just asserting it to be true? $\endgroup$ – Michael McGovern Feb 28 '18 at 4:18
  • $\begingroup$ @MichaelMcGovern: No, that's the induction hypothesis. $\endgroup$ – Eric Wofsey Feb 28 '18 at 4:23
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You have the right idea, but some of your notation and phrasing is off.

First, note that a statement like "$P(t=1)=n_1$" doesn't make sense, for several reasons. First, you've never defined what $P(t)$ means. Second, even if you had defined it, $P(t=1)$ would not make sense; instead you would just write $P(1)$. Third, the usual thing to call "$P(t)$" here would be the statement that if $n_1,\dots,n_t$ are even integers, then so is their sum. So $P(t)$ (or $P(1)$) is an entire statement, not just a number. It doesn't make sense to say $P(1)=n_1$.

What you can instead say for your base case (after defining what $P(t)$ means) is something like:

To prove $P(1)$, suppose $n_1$ is an even integer. Then the sum we must prove is even is just $n_1$, which we know to be even.

Note how I clearly introduced the variable $n_1$ with "suppose $n_1$ is an even integer" before proceeding to prove anything. This is in general a good practice when proving a "for all" statement: you first start by explicitly mentioning that you are considering a particular instance of the statement.

There are similar issues with your phrasing in the inductive step. Also, the induction hypothesis (where you assume that $P(t)$ is true) should be the start of the inductive step, rather than the end of the base case.

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  • $\begingroup$ Does this look better: Assume P(t), i.e. if n1, n2,...,nt are even, then n1+n2+...+nt is even, is true. Then Since n1+n2+...+nt is assumed to be even and nt+1 is even, an even + even gives an even number. Therefore P(t+1) is shown to be true by induction. $\endgroup$ – paul9797 Feb 28 '18 at 4:28
  • $\begingroup$ Yeah, that's a lot better! My main suggestion would be that you explicitly introduce the variables $n_1,\dots,n_{t+1}$ after your first sentence, like I did for the base case. $\endgroup$ – Eric Wofsey Feb 28 '18 at 4:36

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