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The fact that 1, 2, 3, 4, 5, and 6 are all factors of 60 does not mean that 7 is going to be a factor of On the other hand, 2 times 3 divided by 2 is also 3. Use OCW to guide your own life-long learning, or to teach others. And this is not quite as trivial as it might otherwise have seemed. This becomes 41 squared, minus 41, plus What mathematical induction says is this, let’s suppose we have a conjecture.

And, while you’re thinking about that, let’s take a break for a few more asides which, I think, may cement down this idea a little bit more strongly. Now, you see our previous case told us, what?

The Definite Integral Part V: The sum of the first ‘n’ numbers, no matter how many you have, is, what? And now, what we’ve done is we have verified the second part of our mathematical induction setup, namely, if we go back to our basic definition over here, we have to show, what?

Now, again, this may be a very naive way of looking at it, but I always look at mathematical induction as a bunch of toy soldiers stacked up in a line in such a way that, if any one of the toy soldiers falls down, he knocks down the one that’s immediately behind him, OK?

Consecutive odd numbers, which are both prime, are called twin primes. Let me show you what I mean by that. Now that we have the definition, let’s proceed directly to use it. On the number line, here’s 0. Let’s just call mathematica, ‘n’ equals 7. Mathemqtical this, in turn, says, what?

No, there are problems that do downloaad lend themselves to induction. Now, you see, we’ve reduced our problem to the sum of two functions, namely, ‘f1 of x’ plus ‘f2 of x’ being one of our functions, ‘f3 of x’ being another of our functions. Well, what we must do next is investigate what happens if you add, what?

That’s a nice operation. Up above, we assumed that the sum ldf two functions was a function. We try to reduce an unfamiliar problem to a familiar problem which has already been solved.

Principle of Mathematical Induction NCERT Solutions – Class 11 Maths

In other words, it wasn’t just that we proved that the formula was true for the sum of three functions, we proved it on the assumption that it was already true for the sum of two functions. A ” ” symbol is used to denote such documents. Look at, if ‘n’ is 1, the left-hand side here is 1.

So, 2 is the only even prime. Of course, what we assumed in doing this was that the sum of two numbers was, again, a number. Namely, we learned, what? Now, take ‘k’ to be 1.

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction

And what we said was, look at, we’ll just add these two at a time. This says that every positive whole number greater than one can be factored uniquely into a product of primes, unique up to the order in which you write them. And now I’ll loosely use the word et cetera, and come back and reinforce that as we go along. It’s an anecdote attributed to the mathematician, Gauss, who, when he was a young chap, was a discipline problem in school. If you’d like a more fascinating, realistic example, it’s something that we call the unique factorization theorem of elementary number theory.

And, by the way, one more little aside.

We’ve shown that knowing that the limit of a sum is the sum of the limits was true pf a sum of two functions and of three functions. And the generalization inudction what is known as mathematical induction.

The first one plus the last one added up to And, now, given the limit of the sum of ‘n’ functions, we know that that’s the limit of a sum.

Because, evidently, the fact that this was a prime in no way depended structurally on the fact that this one was a prime.

Let’s see what I mean by that. In fact, in the form of a rather interesting aside, there is a very interesting mathematical anecdote connected with this particular problem. The point that I’m making is why shouldn’t induction have some weaknesses?

They must all be here. You see what I’m driving at here? And, by the way, notice how we may now begin to suspect that this idea generalizes. That is, if we have two functions, ‘f 1’ and ‘f 2’, the limit of ‘f1 of x’ plus ‘f2 of x’ as ‘x’ approaches ‘a’ is the limit of ‘f1 of x’ as ‘x’ approaches ‘a’ plus the limit of ‘f2 of x’ as ‘x’ approaches ‘a’.

And, you see, we know that the limit of a sum is the sum of the limits is true if we have only two functions, so, from here, we get to here. Well, so far, so good.