Middle Terms

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Eleven students stand in a line for the morning assembly. Who stands exactly in the middle? The sixth one, right? Five on each side. Now, x plus a, the whole thing to the power ten, also has eleven terms. So which one is the middle term? Many students say the fifth, because ten over two is five. Is that right? Hold that thought. And with ten students in the line, who is in the middle then? Middle terms come up often in JEE, so let's settle both. Let's begin.

0:37Which term sits in the middle?

So, first, count the terms. x plus a, in round brackets, to the power n, where n is a positive integer, has n plus one terms. We call them T one, T two, and so on up to T n plus one: capital T with the term number written small below it, and the three dots mean and so on. Okay? Now, our puzzle. Here n is ten, so there are eleven terms. Ten over two is five, so the middle term is T five, right? No, no, careful! Count it like the students in the line. Eleven terms, and the double arrow on screen means it follows that the middle term is T six, with five terms on each side. With me? So, when n is even, n plus one is odd, and there is exactly one middle term: T, n over two plus one, with that number written small below the capital T, okay? Ten over two, plus one, is six. Yes, T six! Clear? Now the second half of our puzzle. n equals nine gives ten terms. Ten students in a line: who stands in the middle? Yes, exactly! Nobody stands there alone. The fifth and the sixth share the middle, four on each side. So two middle terms: T five and T six. Got it? So, when n is odd, n plus one is even, and there are two middle terms: T, n plus one over two, and T, n plus three over two. Put n equals nine and you get T five and T six. Right? Now, to write a middle term, use the general term: T r plus one equals n C r, the small n at the top left of the capital C, times x to the power n minus r, times a to the power r. We count from r equals zero, so r is one less than the term number: for T six, r is five. Mixing up r and r plus one is a very common slip here, so be careful. Okay?

2:31The middle term has the greatest binomial coefficient

Now, something special about the middle term. Here are the binomial coefficients for n equals four, four C zero up to four C four: one, four, six, four, one. The biggest, six, sits right in the middle. And see, terms equally far from the two ends have equal binomial coefficients: four and four, one and one. With me? Now n equals five. Where will the biggest one be this time? One, five, ten, ten, five, one. Two equal tens, side by side in the middle, just like our two middle terms. Clear? So, when n is even, the greatest binomial coefficient is n C, n over two: the one in the middle term. For n equals four, that is the six. Right? When n is odd, there are two greatest: n C, n minus one over two, and n C, n plus one over two. They are equal, because they are equally far from the ends: for n equals five, the two tens. Got it? Oh no, careful here! Binomial coefficient means n C r only. The full coefficient of a term also takes in the numbers inside the brackets, and that is a different thing. So read the question carefully. Okay?

3:48Example 1: middle term of (x/3 + 9y)^10

Alright, example one. Find the middle term of: x over three, plus nine y, the whole thing to the power ten. Pause the video and try it. Done? Okay, let's check. n is ten, even, so one middle term: ten over two, plus one, that is T six. So r equals five. Right? T six equals ten C five, times x over three to the power five, times nine y to the power five. Ten C five, the greatest binomial coefficient, sits in the middle term, as promised. With me? Now the numbers. Nine to the power five on top, three to the power five below. What does that simplify to? Correct! Nine over three is three, so it is just three to the power five. Ten C five is two hundred fifty two, and the dot means times. Clear? Three to the power five is two hundred forty three, and the cross sign also means times. Two hundred fifty two times two hundred forty three: sixty one thousand two hundred thirty six. So the middle term is sixty one thousand two hundred thirty six, x to the power five, y to the power five. And it was T six, not T five, just like our puzzle. Got it?

5:03Example 2: two middle terms of (2x - x^2/4)^9

Now a harder one. Find the middle terms of: two x, minus x squared over four, the whole thing to the power nine. Terms, plural. Pause the video and try it. Hint: n is odd here. Done? Okay, let's check. n is nine, odd, so ten terms and two middle terms. Nine plus one over two is five, and nine plus three over two is six. So T five and T six: r equals four, and r equals five. Okay? First, T five. Nine C four, times two x to the power five, times, in round brackets, minus x squared over four, to the power four. I keep the minus sign inside the bracket. Right? Quick question. Minus x squared over four, to the power four: will T five come out positive or negative? Positive, yes! An even power kills the minus sign. Nine C four is one hundred twenty six, two to the power five is thirty two, and four to the power four is two hundred fifty six. Power of x: five plus eight, thirteen. So T five is sixty three over four, x to the power thirteen. Clear? Now T six, with r equals five. Nine C five, times two x to the power four, times minus x squared over four, to the power five. This time the power is odd, so the minus sign stays! And nine C five equals nine C four: the two middle terms share the greatest binomial coefficient. With me? Two to the power four is sixteen, four to the power five is one thousand twenty four, and the power of x is four plus ten, fourteen. So T six is minus sixty three over thirty two, x to the power fourteen. Drop the sign, or give only one middle term, and the marks are gone. Got it?

6:57Quick revision: middle terms for n even and n odd

So, quick revision. One: n even, one middle term, T, n over two plus one. Two: n odd, two middle terms, T, n plus one over two, and T, n plus three over two. Three: put r equals the term number minus one in T r plus one, and keep the signs inside the brackets. Four: the middle term, or the two middle terms, carry the greatest binomial coefficient. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!