Sequences and Series

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Here is a list: one, seven, seventeen, thirty one, and it goes on forever, following one rule. Is one hundred and twenty seven somewhere in this list? Hmm. You could keep writing terms and hope to land on it. But what if the number was in the thousands? Today we learn to answer this in one line. And every later topic of this unit, the progressions and the sums of series, starts from today's ideas. Questions on sequences come up often in JEE, so let's begin.

0:36What is a sequence?

So, what is a sequence? Think of a numbered list: a first term, a second term, a third term. Formally, it is a function whose domain is the natural numbers. The double-lined capital N on screen is that set, written in curly brackets: one, two, three, and the three dots mean and so on. Okay? Now tell me. Can a sequence have a term at position two and a half? No, never! Positions are counted: first, second, third. That is why the domain is the natural numbers. Remember this for Example one. Right? So each natural number n goes to one term. The arrow with a small bar at its tail means: goes to. T n is a capital T with a small n below it, a subscript. T one is the first term, T two the second, and so on. And T n, the term at a general position n, is the general term, or the n-th term. Clear?

1:31Two ways to give a sequence

Now, how can a sequence be given? Way one: a formula for T n. Put n equals one, two, three, and out come the terms. Okay? Way two: a recurrence, which gives each term from the one before. T of n plus one equals T n, plus two n. Here the whole n plus one sits in the subscript, below the T, and it means the next position. Got it? So, suppose T one is one. What is T two? Three, correct! Put n equals one: T two is one plus two times one, three. n equals two: T three is three plus four, seven. n equals three: T four is seven plus six, thirteen. Each term needs the one before it. Right?

2:15Series and the sigma sign

Now, a sequence is a list. When we add its terms, T one plus T two, up to T n, we get a series. The big symbol is sigma, the capital Greek letter, and it means: add up. Here k is a counter. It starts at the value below sigma, one, and stops at the value on top, n. Okay? This sum of the first n terms gets its own name: S n, a capital S with a small n below it. So S one equals T one, just the first term, and S two equals T one plus T two. Clear? Let's read one. Sigma, for k from one to four, of two k minus one. What does it add up to? Put k equals one, two, three, four: one, three, five, seven. Total sixteen. Very good! Sigma is just a short way to write a long sum. Got it?

3:04From the sums back to the terms

Now the most useful idea of today. Suppose a question gives you S n and asks for the terms. How do we go back? Write S n in full: T one plus T two, up to T of n minus one, plus T n. Okay? Below it, S of n minus one; the whole n minus one is the subscript. It is the same sum, stopping one term earlier. Now subtract the second line from the first. What is left? Only T n, exactly! Everything else cancels. So T n equals S n, minus S of n minus one, for n greater than or equal to two. That sign is greater than, with a line below it. Right? But why only from n equals two? What goes wrong at n equals one? At n equals one, you would need S zero, a sum of zero terms, but our n starts from one. So the first term comes straight from the first sum: T one equals S one. Okay? And here is a common trap. Students use the subtraction formula for every n, without checking. Oh no, careful! Put n equals one in your formula. Does it give S one? If yes, one formula works for all n. If not, write T one separately. With me?

4:18Example 1: is 127 a term when Tn = 2n^2 - 1

Now, remember our puzzle? Let's solve it. T n equals two n squared minus one; the small two up high is the square. Write the first four terms. Is one hundred and twenty seven a term? Pause the video and try it. Done? Okay, let's check. n equals one: two minus one, one. n equals two: eight minus one, seven. Then eighteen minus one, seventeen, and thirty two minus one, thirty one. Our puzzle list! Right? Should we keep listing terms until we reach one hundred and twenty seven? There is a faster way. What is it? Yes, exactly! Set T n equal to one hundred and twenty seven, and solve. Two n squared is one hundred and twenty eight, so n squared is sixty four. The double arrow means: this gives. Okay? So, is n eight, or minus eight? The algebra says plus or minus eight; on screen, that is a plus sign sitting on a minus sign. But n is a position, a natural number, remember? There is no term number minus eight! So minus eight is rejected. Clear? So n is eight, and one hundred and twenty seven is T eight, the eighth term. Puzzle solved, in one line! If no natural number n had come out, say only a negative value or a fraction, the number would not be a term at all. Got it?

5:44Example 2: find Tn when Sn = 2^n - 1

Now a slightly harder one. The sum of the first n terms is S n equals two to the power n, minus one; the n up high is the power. Find T n, and name the sequence. Pause the video and try it. Done? Okay, let's check. For n greater than or equal to two, T n equals S n, minus S of n minus one. For S of n minus one, replace n by n minus one: in brackets, two to the power n minus one, minus one. Careful, the whole n minus one is the power. Okay? The minus ones cancel, leaving two to the n, minus two to the power n minus one. Take out two to the power n minus one, and what stays is two minus one, just one. So T n equals two to the power n minus one. Right? Many students stop here. Are we done? No, no, wait! That formula needed n greater than or equal to two. So check n equals one: T one equals S one, two minus one, one. And the formula gives two to the power zero, also one. Same! Clear? So T n is two to the power n minus one for every n: one, two, four, eight, and so on. With me? Each term is two times the one before. Such a sequence is a G P, a geometric progression, and it is coming in the next chapter. Okay? One quick check: S three is one plus two plus four, seven, and two cubed minus one is also seven. Perfect!

7:17Quick revision: Tn = Sn - Sn-1 and the T1 check

So, quick revision. First, a sequence is a list T one, T two, T three, with n always a natural number. Second, a series is a sum, and S n is the sum of the first n terms. Third, T one equals S one, and T n equals S n, minus S of n minus one, for n from two onwards; then check T one. Now go and try the practice problems on this page. Next class: arithmetic progressions. All the best!