Problem 49 Hard Difficulty

Show that $ \frac{1}{2} (T_n + M_n) = T_{2n} $.

Answer

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Video Transcript

Okay, so this question wants us to prove that the average of a trap is oId approximation with the midpoint is a tramp aside approximation with twice as many sub intervals. So there's a geometric argument and in algebraic, but I think it's easier just to work through the algebra for the sword. So let's just compute the left hand side. So we know that T seven is just Delta X over too times half of a plus, two times all the middle values plus F A B. And then the midpoint approximation is Delta X times F of A plus Delta X over too, plus Beth of a plus to Delta X. Sorry plus three Delta X over too plus all the way up. So now let's compute 1/2 times there some so 1/2 of Tien plus Emma. So this is 1/2 times 1/2 Delta Axe times effort Bay plus to half of a plus Delta acts, plus all the way up toe effort be. And then we also have with 1/2 times off the midpoint turnips. So no, to see something interesting that happens. Cancellation wise. So what's really messing us up, right? now is these twos in the middle of the trap is laid approximation. So let's just multiply one of those one half's in everything. So we'll end up kidding the family. So he's still have a 1/2 all the way pulled out. Oh, I missed a factor of Delta X here. Oops. Okay, so we can actually pull out that Delta X over, too. So we get after they over too, plus f of a plus X plus half of a plus to doubt the axe. Plus that will be plus F of a plus Delta X over too plus f of a plus three Delta X over too plus all the way out. So now let's group thes midpoint songs in the order of their ex values. So we get still toe acts over too times f of a over to plus F of a plus Delta X over too plus f of Delta, eh for a plus Delta X plus f of a plus three Delta X over too plus all the way up to our f b Over Too interesting. So now this looks awfully familiar. This looks like a trap. Is oId on the inside because again we have our terms on the outside and then two time's a factor that on the inside. So what does this turn into? Well, this could be transformed as follows. This is just a trap is laid some with to end sub intervals and we will go back and prove this real quick. I'm just gonna make this assertion because, you see, there's twice as many points inside this middle trap a sight, son, because we add them from the midpoint approximation. So for the last thing we're gonna do to really cement this, we're just gonna pull 1/2 out again. So equals Delta X over four times Fok plus two f of A plus Delta X over too plus two half of a plus Delta X plus two f of A plus three Delta X, all the way up toe F A B And this is exactly the form of a trap aside, some can. There we go. We have proven it. So again, there's a lot of algebra here, so let's just recap what happens. So here's the formulas for approximations and we evaluated this left hand signed by adding them together. And then we did some regrouping to turn this into a trap. Is it some by pulling out the correct factors and reordering it? So we get f of A plus f a B plus two times all the points in between. And since we have twice as many and they're spaced out as they're twice as close together, this is the same thing. Is a trap aside, some with to end intervals?