Geometric Means; A.P. and G.P. Together

Transcript

0:00Introduction

Hello everyone! Small puzzle first. Three numbers in a G P add up to fifty six. Subtract one, seven and twenty one from them, in that order, and they form an A P! What are the numbers? Guessing could take forever, right? By the end of this class, you will crack it in a few lines. Today: geometric means, and A P and G P together. Questions like these come up often in JEE, so let's begin!

0:31The geometric mean of two numbers

So, first, the geometric mean. Put one number, capital G, between a and b, so that a, capital G, b is a G P, okay? From our first G P lesson: three nonzero numbers are in G P exactly when the middle one squared equals the product of the other two. So capital G squared equals a times b. With me? Now you tell me. What is capital G itself? The square root of a b, yes! That is the root sign on screen. We take a and b positive, and capital G is the positive square root. This is the geometric mean, the G M, of a and b. Clear? Quick example: the G M of two and eighteen. Two times eighteen is thirty six, and root thirty six is six. Two, six, eighteen: times three each time, a G P! Perfect!

1:29Inserting n geometric means between a and b

Now, n means. Put n numbers between a and b so that the whole list is a G P, okay? These are capital G with a small number below: G one, G two, up to G n. The three dots mean and so on. With a and b, that makes n plus two terms. Right? So here is my question. Which term is b, and what power of the common ratio r takes you from a to b? Term number n plus two, so the power is n plus one, correct! So b equals a times r to the power, n plus one. The double arrow means it follows that: r equals b over a, in round brackets, to the power one over, n plus one. And the k th mean, G with a small k below, is a times r to the power k. Got it? Let's try one from the course: three geometric means between one and two hundred and fifty six. Three means, so r cubed equals two hundred and fifty six, right? No, no, careful! Same trap as with A Ps. Three means make four gaps: n plus one, remember? So r to the power four is two hundred and fifty six, and r is four. Clear? So the means, in bold, are four, sixteen and sixty four. That's it!

2:52The product of the n means

Now, the product of the means. G one is a times r, one step after a. G n is b over r, one step before b. Multiply: the r's cancel, leaving a times b. Same for G two times G n minus one, and so on: each pair, front with back, gives a b. With me? So, what is the product of all n means? Yes, exactly! The dots between the G's mean times, and so on. All the pairs give a b to the power n over two: root a b, in brackets, to the power n. So the product is the single G M, to the power n. Right? Now one more from the course: the product of five geometric means between two and eighteen. Do we even need the common ratio? No, we don't! Root of two times eighteen is root thirty six, which is six, our G M from before. So the product is six to the power five: seven thousand seven hundred and seventy six. No r at all!

3:55Two numbers from their A.M. and G.M.

One more result, and it is really useful. Suppose you know only the A M, capital A, and the G M, capital G, of two numbers. Let's get the numbers back. The A M: a plus b, over two, equals capital A, so a plus b equals two capital A, okay? The G M: root a b equals capital G, so a times b equals capital G squared. Now we know the sum and the product. Right? So, which equation has two roots with this sum and this product? The quadratic, exactly! x squared, minus the sum times x, plus the product, equals zero. So a and b are the roots of x squared minus two capital A x, plus capital G squared, equals zero. Got it?

4:43Example 1: two numbers with AM 10 and GM 8

Alright, example one. The A M and the G M of two positive numbers are ten and eight. Find the numbers. Pause the video and try it. Done? Okay, let's check. Capital A is ten and capital G is eight, so two capital A is twenty, and capital G squared is sixty four. The equation: x squared minus twenty x, plus sixty four, equals zero. Okay? Now, which two numbers add up to twenty and multiply to sixty four? Four and sixteen, correct! x minus four, times x minus sixteen, equals zero. So the numbers are four and sixteen. Clear? Always check. Four plus sixteen, over two, is ten, and the root of four times sixteen is root sixty four, eight. Perfect!

5:37Example 2: GP with sum 56 that turns into an AP

Now, our puzzle from the start. Three numbers in G P add up to fifty six. If one, seven and twenty one are subtracted from them respectively, that is, in that order, the results are in A P, okay? Find the numbers. Pause the video and try it. Done? Okay, let's check. Take the numbers as a, a r, a r squared. Their sum: a times, in brackets, one plus r plus r squared, equals fifty six. With me? After subtracting: a minus one, a r minus seven, a r squared minus twenty one, and these are in A P, right? So what is the condition for three numbers in A P? Twice the middle equals the sum of the ends, yes! Two times, a r minus seven, equals a minus one, plus a r squared minus twenty one. Simplify: a times, r minus one, whole squared, equals eight. Okay? Now the key move: divide the first equation by this one, and a cancels. One plus r plus r squared, over r minus one whole squared, equals fifty six over eight, which is seven. Cross multiply: six r squared minus fifteen r plus six equals zero. Clear? Divide by three: two r squared minus five r plus two equals zero. So r is two, or r is one half. With me? Now, r equals two looks nice, so we throw away one half, right? Oh no, careful! Both work. From a times, r minus one, whole squared, equals eight: r equals two gives a equals eight, so eight, sixteen, thirty two. And r equals one half gives a equals thirty two, shown in brackets: thirty two, sixteen, eight. The same numbers reversed, okay? Check eight, sixteen, thirty two: subtract one, seven and twenty one, and you get seven, nine, eleven. Up by two each time, an A P! Puzzle solved.

7:44Quick revision: GM, n GMs, numbers from AM and GM

So, quick revision. The G M of positive a and b is root a b. For n geometric means, r is b over a, to the power one over, n plus one, and their product is root a b, to the power n. From the A M, capital A, and the G M, capital G, the numbers are the roots of x squared minus two capital A x, plus capital G squared, equals zero. And for A P with G P, take a, a r, a r squared, and use twice the middle equals the sum of the ends. Now go and try the practice problems on this page. All the best!