Prove the following proposition using the principle of mathematical induction

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Prove the following proposition using the principle of mathematical induction

Mathematics
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\[\large\rm \sum_{i=1}^n (2i-1)=n^2,\qquad\forall n\in\mathbb Z^+\] Ok so we start with our base case: \(\large\rm n=1\)
\[\large\rm (2\cdot 1-1)=2-1=1\]Which is equal to \(\large\rm 1^2\), so our base case is satisfied, ya?

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yes
We'll assume it's true for \(\large\rm n=k\). Which means we assume \(\large\rm \color{orangered}{1+3+...+(2k-1)=k^2}\) is true.
We call this the Induction Hypothesis ^
yeah
okay
Then for our Induction Step: \(\large\rm n=k+1\) Let's see if we can get our formula set up correctly :d
to equal n^2?
Well, notice that our last number in the sum will be \(\large\rm k+1\) So we won't end up with \(\large\rm n^2\), we'll end up with \(\large\rm (k+1)^2\) on the right. But setting up the left side is going to be a little tricky. Hopefully you can follow what I'm doing here.
is it not k+2?
Your summation is giving you a bunch of terms like this:\[\large\rm (2\cdot\color{royalblue}{1}-1)+(2\cdot\color{royalblue}{2}-1)+...+(2\color{royalblue}{k}-1)+(2\color{royalblue}{(k+1)}-1)=(k+1)^2\]
I'll simplify the first few terms like I did the last time.\[\large\rm 1+3+...+(2k-1)+(2(k+1)-1)=(k+1)^2\]
k+2? what? :o
i dont get the part (2(k+1)−1)
i dont get where k+1 came from
wait i get it
i get it
XD
So in the Induction Hypothesis, we let \(\large\rm k\) be the largest number in the sequence. We replaced n by k. In our Induction Step, we're letting \(\large\rm k+1\) be the largest number in the sequence. It's the number that comes after k. So our summation grows a little bit, by one more term. And it should be equivalent to (k+1)^2 now, instead of k^2.
yup
If you scroll up, you'll notice I colored our Induction Hypothesis in orange. We need to relate this back to that orange thing.
1+3+...+(2k−1)=k^2
this?
Yes, that entire left side is contained within our Induction Step formula.
\[\large\rm \color{orangered}{1+3+...+(2k-1)}+(2(k+1)-1)=(k+1)^2\]Do you see it? :)
yeah
Since our Induction Hypothesis tells us that the orange stuff is k^2, we can make the switch,\[\large\rm \color{orangered}{k^2}+(2(k+1)-1)=(k+1)^2\]
ohhhhh oh kay
From there, you need to turn the left side into (k+1)^2 somehow. Do some Algebra steps expand out the stuff, then try to factor it down.
thank yyou!
np \c:/

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