Harmonic Progressions

Transcript

0:00Introduction

Hello everyone! Okay, let's start with a small puzzle. Here is a sequence: one half, one fifth, one eighth, and so on. Is it an A P? No, the gaps are not equal. Is it a G P? No, the ratios are not equal either. So, how will you find its tenth term? Tough, right? Today, one simple idea cracks it: the harmonic progression. It comes up often in JEE, usually mixed with A P and G P questions. So, let's begin.

0:33What is an H.P.?

So, here is our sequence. Each term is one over a number, and the three dots mean it goes on. Not an A P, and not a G P, right? Now think. What happens if you flip every term upside down? Two, five, eight! Yes, exactly! The gaps are three and three. So the flipped sequence is an A P with common difference d equal to three. Got it? And that is the whole definition. Numbers a one, a two, a three, and so on, are in harmonic progression, H P for short, when their reciprocals, one over a one, one over a two, and so on, are in A P, okay? Here a one is a small a with a one written below it, and the double arrow pointing both ways means if and only if. Clear? Now, one condition. We flip every term, and one over zero is not defined. So every term of an H P must be non-zero. Right? Now, a tempting shortcut. To add one half, one fifth and one eighth, can we just add two, five and eight, and flip the total? No, no, wait! The slashed equals sign means not equal. Two plus five plus eight is fifteen, and one over fifteen is smaller than one half alone, so it cannot be the sum! In fact, an H P has no sum formula at all. Okay? So, the rule for every H P problem: turn it into an A P, do the work there, then flip back. With me?

2:00Three numbers in H.P.

Now, three numbers a, b, c, in this order, are in H P when one over a, one over b, one over c are in A P, okay? Quick question. For three numbers in A P, what is the test? Twice the middle equals the sum of the ends, very good! So two over b equals one over a plus one over c, which adds up to a plus c over a c. Right? Flip both sides and multiply by two. So, b equals two a c over a plus c: the H P condition on the middle term. Write it down. Got it? One more form, useful in proofs. Equal gaps: one over b minus one over a equals one over c minus one over b. Simplify each side; the arrow means implies: a minus b over a b equals b minus c over b c. Clear? Cross multiply and cancel b. So, a minus b over b minus c equals a over c. Three forms, one idea, okay?

3:02The harmonic mean of two numbers

Now, the harmonic mean, H M for short. If a, capital H, b are in H P, then capital H is the H M of a and b, okay? Our middle term formula gives it at once: capital H equals two a b over a plus b. Right? Here is the topper's way to remember it: flip, average, flip back. One over capital H is half of, one over a plus one over b; the big round brackets hold that sum. Clear? Now, a quick one. The H M of four and twelve. Their average is eight, so the H M is one over eight, right? No, no, careful! That flips the average of the numbers, not the average of the flips. The dot means times: two times four times twelve, over four plus twelve. Ninety six over sixteen: six, not one eighth! Got it? Let's check. One quarter, one sixth, one twelfth: in twelfths, three, two, one. Equal steps of minus one twelfth, so four, six, twelve really are in H P, okay?

4:05Inserting harmonic means

Next, inserting harmonic means. For n harmonic means between a and b, insert n arithmetic means between one over a and one over b, then flip them all back. Simple, right? Let's try three harmonic means between one half and one sixth. First, flip the ends: two and six. Now, which three arithmetic means fit between two and six? Three, four, five, correct! Five terms, four equal gaps, so d is one. With me? Now flip back: one third, one quarter and one fifth. Students often stop at three, four, five. Don't forget the last flip! Clear?

4:45Example 1: 10th term of the HP 1/2, 1/5, 1/8, ...

Now, remember our puzzle? Let's finish it. Find the tenth term of the H P: one half, one fifth, one eighth, and so on. Pause the video, and try it first. Done? Okay, let's check. Flip every term: two, five, eight, an A P with first term a equal to two and common difference d equal to three. Right? Now the tenth term of this A P, written T ten: a capital T with a small ten below it. T ten equals a plus nine d. Two plus nine times three: twenty nine. So the answer is twenty nine, right? Oh no, careful! That is the tenth term of the A P, not of the H P, okay? Flip it back: the tenth term of the H P is one over twenty nine. Our puzzle, solved in seconds! With me?

5:34Example 2: HP with mth term n, nth term m, find (m+n)th

Now a harder one, with letters. The m-th term of an H P is n, and its n-th term is m, where m is not equal to n. Find its m plus n-th term; the m plus n sits in round brackets on screen. Pause the video and try it. Hint: flip everything first. Done? Okay, let's check. Flip to the A P of reciprocals, with first term capital A and common difference capital D, okay? Its m-th term, capital A plus, m minus one in round brackets, times capital D, equals one over n. Its n-th term equals one over m, in the same way. Right? Two equations, two unknowns. How do we get rid of capital A? Subtract, exactly! We get m minus n, times capital D, equals one over n minus one over m, which is m minus n over m n. We can cancel m minus n only because m is not equal to n. That is why the question says so! So capital D equals one over m n. Clear? Put this back in the first equation. Capital A equals one over n, minus, m minus one, over m n. That simplifies to one over m n. So capital A and capital D are equal. Nice, right? Now the m plus n-th term of the A P: capital A plus, m plus n minus one, times capital D, which is m plus n over m n. Okay? And the last step toppers never forget: flip back! The m plus n-th term of the H P is m n over m plus n. Flip, work in the A P, flip back. That's it!

7:17Quick revision: flip to an AP, HM = 2ab/(a + b)

So, quick revision. In an H P, the reciprocals are in A P, every term is non-zero, and there is no sum formula. For three numbers in H P, b equals two a c over a plus c, and the H M of a and b is two a b over a plus b. For every problem: flip, work in the A P, flip back. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!