The General Term of an A.P.

Transcript

0:00Introduction

Hello everyone! First, a small puzzle. You are standing on the first step of a staircase, and you want to reach the twentieth step. How many steps do you climb? Twenty, right? Hmm, are you sure? Today we learn the general term of an arithmetic progression, and our staircase is hiding inside it. This formula is the base of the whole unit on sequences and series, and it comes up often in JEE questions. So, let's begin.

0:30What is an A.P.?

So, what is an arithmetic progression, an A P for short? It is a sequence in which each term, minus the term before it, is always the same constant. Start with the first term, small a, and keep adding the same number d: a, a plus d, a plus two d, a plus three d. The three dots mean it goes on. Okay? That constant d is the common difference. T stands for term, and the small number below it is a subscript: T one is the first term, T two the second. So d equals T two minus T one, equals T three minus T two, and so on. With me?

1:07The general term

Now, let's write the terms one by one. T one is a. T two is a plus d. T three is a plus two d. T four is a plus three d. Look at the pattern: the number of d's is one less than the position of the term, right? So, tell me. How many d's will the twentieth term have? Nineteen, yes, exactly! T twenty equals a plus nineteen d. And this is our staircase! From the first step to the twentieth, you climb only nineteen steps, not twenty. In the same way, you add d only nineteen times. Clear? So, in general, T n equals a, plus n minus one, in brackets, times d. This is the general term, the n-th term. Write it down. And the classic mistake? Writing a plus n d. That is one step too far! Always n minus one, okay?

2:00Counting from the end

Now, sometimes the A P stops at a last term: small l, the letter, not the number one. Then the n-th term from the end equals l, minus n minus one, in brackets, times d. Right? Look, read the A P backwards. You start at l, and every step back takes away d. So it is again an A P, with first term l and common difference minus d. Same staircase, just walking down. Got it?

2:27Properties of an A.P.

Now some properties that save time. Add the same constant to every term, or subtract it: the differences do not change. Multiply every term by the same constant, or divide by the same non-zero constant: the common difference is multiplied or divided too. Either way, you still have an A P, okay? Next, a quick test. You are given three numbers, a, b and c, in this order. How do you check that they are in an A P? Yes, the two differences must be equal: b minus a equals c minus b. Rearrange it: two b equals a plus c. The double arrow on screen means if and only if: the test works both ways. So the middle number is exactly halfway between the other two. Simple, right? Now take a finite A P with n terms. Here, a with a small subscript means a term: a one is the first, a n the last, a k the k-th. Then a k, plus a sub n minus k plus one, equals a one plus a n. So terms at the same distance from the two ends add up to first plus last. Step in from the left, you gain d; from the right, you lose d. Clear? And more generally, for any A P: a p plus a q equals a r plus a s, whenever p plus q equals r plus s. Only the positions matter. Got it?

3:51Topper's trick: choose the terms symmetrically

Now a trick that toppers use. When three numbers in A P have a given sum, do not take a, a plus d, a plus two d. Take a minus d, a, a plus d, with a in the middle. Now tell me, what is their sum? Three a, exactly! The minus d and the plus d cancel. So the sum gives you the middle term straight away. One equation, one unknown, done. Nice, right? For four terms, we want the same balance. So take a minus three d, a minus d, a plus d, a plus three d. But look carefully. What is the common difference of these four terms? Two d, not d! From a minus three d to a minus d, you add two d. And the sum is four a; the d's cancel again. Students often report d as the common difference. Here it is two d. Careful, okay?

4:43Example 1: 5th term from the end of 2, 5, 8, ..., 62

Okay, Example one, an easy one, from the course. Find the fifth term from the end of the A P two, five, eight, and so on, up to sixty two. Pause the video, and try it first. Done? Okay, let's check. The last term, l, is sixty two. And d is five minus two, that is three, okay? So, the fifth from the end is sixty two minus five times three, that is sixty two minus fifteen, forty seven. Right? No, no, careful! Remember the staircase. The last term itself is the first from the end. So the fifth from the end is only four steps back: l, minus five minus one, in brackets, times d, that is l minus four d. With me? So, sixty two minus four times three, that is sixty two minus twelve: fifty. Let's be sure. Count back in threes: sixty two, fifty nine, fifty six, fifty three, fifty. The fifth one is fifty. Perfect!

5:43Example 2: three numbers in AP, sum 27, product 504

Now Example two, a little harder, also from the course. Three numbers in A P have sum twenty seven and product five hundred and four. Find them. Hint: use the topper's trick. Pause the video, and try it. Done? Let's check. Take the numbers as a minus d, a, a plus d. Their sum is three a, so three a equals twenty seven, and the arrow means this gives a equals nine. One line, and we have the middle number! Now the product. Nine minus d, times nine, times nine plus d, equals five hundred and four. The dot means times, okay? Nine minus d, times nine plus d, is eighty one minus d squared; the small two up there means squared. So nine times, eighty one minus d squared, is five hundred and four. Divide by nine: eighty one minus d squared equals fifty six. So d squared is twenty five, and d is five. Right? Oh no, careful! d squared equals twenty five gives two values. d equals plus or minus five; that plus-minus sign means both. Never drop one of them, okay? With d equal to five: four, nine, fourteen. With d equal to minus five: fourteen, nine, four. The same three numbers, in reverse order! Quick check. Four plus nine plus fourteen is twenty seven, and four times nine times fourteen is five hundred and four. That's it!

7:13Quick revision: nth term, term from end, a-d, a, a+d

So, quick revision. The n-th term is a, plus n minus one times d; from the end, l, minus n minus one times d. Always n minus one, like the staircase. Three numbers a, b, c are in A P if and only if two b equals a plus c; and a p plus a q equals a r plus a s when p plus q equals r plus s. And choose three terms as a minus d, a, a plus d, or four with common difference two d. Now try the practice problems on this page. All the best!