Overview

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Write the counting numbers, but skip every perfect square: two, three, five, six, seven, eight, ten, and so on. One, four and nine are gone, right? Now, which number sits at position two thousand and twenty six in this list? Writing them all out would take hours. Hold that thought; we will settle it in Example two. Sequences and series comes up often in JEE, and this chapter, the unit problem bank, is where you test yourself on the whole unit. Today I will show you how to use it, and then we will solve two of its problems together. So, let's begin.

0:46Part A: a timed test

So, let's open this chapter and see how to use it. First, Part A: eight solved problems at JEE Main level. Okay? And they are mixed from the whole unit: sums of an A P and a G P, the means, telescoping sums, and sums like sigma of, k squared plus one, in round brackets. The big Greek letter sigma means, add them all up. Right? Now, the course is very clear: attempt Part A as a thirty minute test, after you finish Chapters one to five. So first the chapters, then sit down with a clock, just like the real exam. Clear? Quick question for you. Eight problems, thirty minutes. How much time can you give each one? Thirty over eight: three point seven five. So a little under four minutes a problem, yes, exactly! Read it, spot the idea, solve it, and move on. Got it?

1:43Part B: JEE Advanced level

Next, Part B: five longer problems, at JEE Advanced level. With me? Here is the kind of problem you will meet. An identity about an A P that you must prove, and the sum of k times two to the power k, where the dot means times. Okay? Then the sum of all three digit numbers whose digits are all odd, and the two thousand and twenty sixth non-square, yes, our puzzle! Right? And the least value of the product of one plus a, one plus b, and one plus c, each in round brackets, for a, b, c greater than zero, with a b c equal to one. Each of these joins two or three ideas from the unit, so give them time. Clear?

2:26Part C: concept checks

And finally, Part C: five concept checks. They test the ideas themselves, the kind of thing that is often tested in statement form: true or false. Okay? Here is one of those concept checks. True or false: an infinite G P, with common ratio r equal to minus one, has a sum. Many students put r equals minus one into a over one minus r, and get a over two. So, true? No, no, careful! Let's actually add it. With r equal to minus one, the terms are a, minus a, a, minus a, and so on. Add them one by one, and the partial sums go a, zero, a, zero, again and again. With me? They never settle on one value, so there is no sum. False! The formula, S infinity equals a over one minus r, needs mod r less than one. That is S with the infinity sign below it, and the two vertical bars around r mean its size, its modulus. So check the condition first, every single time. Remember this one!

3:34Example 1: sum of 4 - 2 + 1 - 1/2 + ... to infinity

Alright, here is an easy one from Part A, to warm up. Find four, minus two, plus one, minus one half, and so on, to infinity. The three dots mean the pattern goes on forever. Pause the video, and try it. Done? Okay, let's check. The first term, a, is four. The common ratio, r, is the second term over the first: minus two over four, which is minus one half. So the signs keep switching. Okay? Now, remember the lesson from Part C: check the condition first. Mod r is one half, which is less than one, so the sum exists. Right? Now, many students write four over one minus one half, and get eight. Quick and easy. Is that right? No, no, careful! The ratio is minus one half, not one half. So S infinity is four over one minus, minus one half, in round brackets. Minus a minus becomes plus: four over one plus one half. With me? One plus one half is three over two. So we have four over three halves, which is eight over three. The answer is eight over three. Very good!

4:51Example 2: the 2026th positive integer that is not a square

Now our puzzle from the start, as a Part B problem. Read it with me: the positive integers that are not perfect squares, in increasing order, are two, three, five, six, seven, eight, ten, and so on. Find the two thousand and twenty sixth of them. This one needs a little thinking, so take your time. Pause the video, and try it. Done? Okay, let's check. Let's call the answer capital N, okay? Every number up to N is either a square or a non-square. So N, minus the number of squares up to N, in round brackets, must be two thousand and twenty six. Clear? So the real question is this: about how many squares do we skip on the way? About forty five, yes! Because forty five squared is two thousand and twenty five. So try N equals two thousand and twenty six, plus forty five, which is two thousand and seventy one. Okay? So the answer is two thousand and seventy one, right? Hmm, wait. We jumped forward by forty five numbers. Could a new square sneak in on the way? Very good question to ask! The squares up to two thousand and seventy one are one squared, two squared, and so on, up to forty five squared. Forty six squared is two thousand one hundred and sixteen, too big. So no new square sneaks in: exactly forty five squares. With me? And two thousand and seventy one is not a square itself. So two thousand and seventy one, minus forty five, is two thousand and twenty six. It fits perfectly. Got it? So the two thousand and twenty sixth non-square is two thousand and seventy one. Puzzle solved, and no hours of writing! Beautiful, isn't it?

6:39Quick revision: infinite GP sum and counting non-squares

So, quick revision. Part A as a thirty minute test, Part B at JEE Advanced level, and Part C, the concept checks. S infinity equals a over one minus r, only when mod r is less than one, and watch the sign of r. With r equal to minus one, the partial sums swing between a and zero, so there is no sum. And for non-squares: capital N, minus the squares up to N, equals two thousand and twenty six; then check the next square. Now set your thirty minute timer, start Part A, and go. Then try the practice problems on each page. You can do this, I promise. All the best!