Consecutive Coefficients

Transcript

0:00Introduction

Hello everyone! Okay, look at three numbers: one thousand and one, two thousand and two, three thousand and three. They are in the ratio one is to two is to three. And they sit side by side, as neighbouring coefficients, in one binomial expansion. If I give you only the ratio, can you find the power n, and the place? Sounds hard, right? By the end of today, it takes a few lines. Consecutive coefficients come up often in JEE, so let's begin.

0:32First decide: which three coefficients?

So, we work with one plus x, in round brackets, to the power n, where n is a positive integer. The general term is T r plus one, a capital T with a small r plus one below. It equals n C r, the small n at the top left of capital C, times x to the power r, counting from r equals zero. So the coefficient of x to the power r is n C r. Okay? Now a quick question. Take the r-th term: the r-th, not the r plus one th. Is its coefficient n C r? No, no, careful! The r-th term comes one place before T r plus one, so the index is one less: its coefficient is n C r minus one. So the r-th, r plus one th and r plus two th terms have coefficients n C r minus one, n C r, and n C r plus one: three neighbours. Right? But if the question says the coefficients of x to the power r, r plus one, and r plus two, then they are n C r, n C r plus one, and n C r plus two. With me? Even a shift of one changes every answer. So, before any algebra, decide which three the question means. Clear?

1:50Ratio of neighbours: two equations

Now the main tool: the ratio of neighbours. n C r over n C r minus one equals n minus r plus one, over r. Top minus bottom plus one, over bottom. Okay? So tell me: what is the next ratio, n C r plus one over n C r? Yes, exactly! Just put r plus one in place of r. Top minus bottom plus one: n minus r minus one plus one, that is n minus r, over r plus one. Got it? Now the whole method. Suppose the three neighbours are in the ratio a is to b is to c; the colons mean is to. The double arrow means this gives: n minus r plus one over r equals b over a, and n minus r over r plus one equals c over b. Two equations, two unknowns, n and r. Done! Right? And for one plus k x, to the power n? Each coefficient, small c with a small r below, is n C r times k to the power r. So c r over c r minus one picks up one extra k: n minus r plus one over r, times k. Multiply each ratio by k; forgetting it is a common mistake. Clear?

3:06Progressions: A.P. possible, G.P. and H.P. never

Next, progressions. Three neighbours are in A P, arithmetic progression, when the outer two add up to twice the middle: n C r minus one plus n C r plus one equals two times n C r. Now divide by n C r. Okay? We get neighbour ratios: r over n minus r plus one, plus n minus r over r plus one, equals two. Clear the denominators and simplify: n minus two r, in brackets, whole squared, equals n plus two. Write this condition down. With me? Now think. Can three neighbouring coefficients be in G P, geometric progression? Let's test it. In a geometric progression the ratio stays the same, so n minus r plus one over r equals n minus r over r plus one. Cross multiply and simplify: n plus one equals zero. But n is a positive integer! Impossible. Right? So, never in a geometric progression, and never in H P, harmonic progression, either. If an option says otherwise, strike it out. Got it? One ready result for four in a row: a one to a four, with small subscripts, four consecutive coefficients of one plus x to the power n. Then a one over a one plus a two, a two over a two plus a three, and a three over a three plus a four are in arithmetic progression. Okay?

4:36Example 1: coefficients in ratio 1:2:3, find n and r

Alright, example one. The coefficients of the r-th, r plus one th and r plus two th terms of one plus x, to the power n, are in the ratio one is to two is to three. Find n and r. Pause the video and try it. Done? Okay, let's check. First, which coefficients? These terms give n C r minus one, n C r and n C r plus one, in the ratio one is to two is to three: our three neighbours. Right? First ratio: n minus r plus one over r equals two over one. So n minus r plus one equals two r, which gives n equals three r minus one. With me? Now the second ratio, n minus r over r plus one. What does it equal? Three over two, correct! Cross multiply: two n minus two r equals three r plus three, so two n equals five r plus three. Okay? Now put n equals three r minus one into it: two times, in brackets, three r minus one, is six r minus two, which equals five r plus three. So r equals five, and n equals three times five minus one, that is fourteen. Very good! Always check. Fourteen C four is one thousand and one, fourteen C five is two thousand and two, fourteen C six is three thousand and three. Our puzzle numbers! Mystery solved: they are neighbours in one plus x, to the power fourteen. Perfect!

6:13Example 2: (1 + 3x)^n coefficients in ratio 1:8:42

Now a harder one. Three consecutive coefficients of one plus three x, to the power n, are in the ratio one is to eight is to forty two. Find n. Pause the video and try it. Done? Okay. First ratio: n minus r plus one over r equals eight, right? No, no, wait! That is the trap. Here k is three. The coefficients are c r minus one, c r and c r plus one, where c r equals n C r times three to the power r. Okay? So the first ratio picks up a three: three times n minus r plus one, over r, equals eight. So three n plus three equals eleven r. Clear? Second ratio: three times n minus r, over r plus one, equals forty two over eight, which is twenty one over four. Cross multiply and divide by three: four n equals eleven r plus seven. With me? Both equations have eleven r. What is the quickest way to remove it? Subtract, yes! Four n, minus the bracket three n plus three, equals seven. So n minus three equals seven, and n equals ten. Correct! And r? From three n plus three equals eleven r, thirty three equals eleven r, so r equals three. So n is ten. One missing three would have spoiled everything, so always multiply each ratio by k. Got it?

7:51Quick revision: ratio equations, A.P. test, never G.P.

So, quick revision. One: the r-th, r plus one th and r plus two th terms give n C r minus one, n C r, n C r plus one. Two: a ratio a is to b is to c gives two equations; for one plus k x, multiply by k. Three: arithmetic progression if and only if n minus two r, whole squared, equals n plus two; never geometric, never harmonic. Now go and try the practice problems on this page. All the best!