Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Okay, a quick puzzle. Expand one plus two x, to the power ten. Which power of x gets the biggest coefficient? The middle term is x to the power five, and the middle binomial coefficient is the biggest. So, x to the power five, right? Hold that thought. Today is Part C of our problem bank: ten statements that sound correct. Is each one exactly right? Statements like these come up often in JEE, in true or false form. So, let's begin.

0:34How many terms?

So, statement one. x plus a, to the power n, with n a positive integer. Does it have n terms? No, n plus one! The general term is T r plus one: capital T, with r plus one written small below. And r runs zero, one, and so on up to n; the three dots mean and so on. Okay? Now x plus y plus z, to the power n. Three letters, so three to the power n terms? No! It has n plus one, times n plus two, over two terms. Three to the power n comes from putting x, y and z all equal to one: it is the sum of the coefficients. Clear?

1:15Naming a term, and equal coefficients

Next. n C r is the C with a small n at the top left. Is the r-th term of x plus a, to the power n, equal to n C r, x to the power n minus r, a to the power r? Oh no, careful! That is T r plus one, because the counting starts at r equals zero. For the r-th term, replace r by r minus one: n C r minus one, x to the power n minus r plus one, a to the power r minus one. Got it? Now, the double arrow on screen means, so. If n C x equals n C y, must x equal y? Not necessarily! n C r equals n C n minus r. So either x equals y, or x plus y equals n. It comes back in Example two. With me?

2:06Two sums of coefficients

Now sums. The big Greek letter sigma means add, from r equals zero up to n. Is the sum of r times n C r equal to n times two to the power n? No: n times two to the power n minus one. Check n equals two: zero plus two plus two is four, and two times two to the power one is four; the dot means times. Right? Next, the sum of minus one to the power r, times n C r. The sign on screen means greater than or equal to. Is this sum zero for every n from zero up? Not quite! The sum is one minus one, to the power n: zero only when n is at least one. For n equals zero there is one term, zero C zero, which is one. Okay?

2:54Coefficients in G.P., and a fractional index

Now a G P, a geometric progression. Can three consecutive binomial coefficients, n C r minus one, n C r and n C r plus one, be in G P? No, never! In a G P, neighbours have equal ratios: n minus r plus one, over r, and n minus r, over r plus one. Cross multiply, and everything cancels except n plus one equals zero. Impossible for a positive integer n. Clear? One plus x, to the power half, equals one, plus x over two, minus x squared over eight, and so on. Can we use it at x equals three? No! The index, one half, is not an integer, so the series is valid only for mod x less than one; the vertical bars mean mod x. Three is far outside. Got it?

3:43Middle term, or greatest term?

Now, size. Is the middle term always the numerically greatest term? Let's check with a real one. The chart shows the eleven terms of three plus two x, to the power ten, at x equals one, as bars: T one up to T eleven. The middle term is T six. But the tallest bar, in orange, is T five: ten C four, times three to the power six, times two to the power four. So no! At x equals one, T five beats the middle term. Okay? Why? Look at the right side of the chart. T r plus one over T r equals eleven minus r, over r, times two over three. This ratio is greater than or equal to one while r is less than or equal to four point four. So the terms rise up to T five, and fall after it. That is what the caption below says. With me? So the middle term only has the greatest binomial coefficient, n C r. Clear? But the greatest term also depends on x and a, the two parts of the bracket. Change them, and the peak moves. Got it?

4:51Our puzzle: the greatest coefficient

Now, our puzzle from the start. One plus two x, to the power ten. Is the coefficient of the middle term, x to the power five, the greatest? No! Here each coefficient carries a power of two as well, and that pushes the peak to the right. x to the power five gets ten C five, times two to the power five: eight thousand and sixty four. x to the power seven gets ten C seven, times two to the power seven: fifteen thousand three hundred and sixty, the greatest. Puzzle solved! Right?

5:25Example 1: term independent of x in (2x^2 - 1/x)^6

Now Example one, an easy one from Part A of this chapter. Find the term independent of x in two x squared minus one over x, all to the power six. Pause the video and try it. Done? Okay, let's check. T r plus one is six C r, times two x squared to the power six minus r, times minus one over x, to the power r. The power of x: two times six minus r, minus r. That is twelve minus three r. Okay? No x left means the power is zero. That gives r equals four, so the answer is the fourth term, right? No, no, careful! That was our third statement, about the r-th term. Twelve minus three r equals zero gives r equals four, and that is T r plus one, T five, the fifth term. Clear? Now the value. Six C four is fifteen, two squared is four, and minus one to the power four is plus one. Fifteen times four: sixty. Perfect!

6:26Example 2: equal coefficients in (1 + x)^43, find r

Now Example two, a little harder, also from Part A of this chapter. In one plus x, to the power forty three, term number two r plus one and term number r plus two have equal coefficients. Find the value of r. Pause and try it. Done? Let's check. In T r plus one, the coefficient is n C r: the term number minus one. So T two r plus one has forty three C two r, and T r plus two has forty three C r plus one. Okay? The coefficients are equal. So two r equals r plus one, and that is all, right? Not all! Our other trap. n C x equals n C y means x equals y, or x plus y equals n. So two r equals r plus one, or two r plus r plus one equals forty three. With me? First case: r equals one. Then both term numbers are three: the same term, T three. That is the trivial answer. Got it? Second case: three r plus one equals forty three, so r equals fourteen. Then forty three C twenty eight equals forty three C fifteen, since twenty eight plus fifteen is forty three: T twenty nine and T sixteen. So r equals fourteen. Perfect!

7:44Quick revision: number of terms, nCx = nCy, middle term

So, quick revision. One: x plus a, to the power n, has n plus one terms, and n C r, x to the power n minus r, a to the power r, is T r plus one. Two: n C x equals n C y means x equals y, or x plus y equals n; and the sum of r times n C r is n times two to the power n minus one. Three: the middle term has the greatest binomial coefficient, but it is not always the greatest term. Four: an index that is not an integer needs mod x less than one. Now try the fourteen practice problems on this page. All the best!