Sums of Powers of Natural Numbers

Transcript

0:00Introduction

Hello everyone! Let's start with a small puzzle. Add one cubed, two cubed and three cubed. One plus eight plus twenty seven: thirty six. Now add just one, two and three. That is six. And six squared? Thirty six again! Add four cubed as well, and you get one hundred. That is ten squared, and ten is one plus two plus three plus four! So, is this a coincidence? Hmm, not at all! Today we will see the formula behind it, and learn three formulas that add up a whole series in one line. Sums like these come up often in JEE, so let's begin!

0:43The sigma sign and the first sum

So, first, a short way to write a long sum. This big Greek letter is capital sigma, and it means add up. Below it, k equals one, and on top, n. So k takes the values one, two, three, up to n, and we add. The three dots mean and so on. Okay? Now, you have met this sum before. One plus two plus three, up to n. What does it add up to? n times n plus one, over two. Yes, exactly! It is an A P with first term one and last term n, so it is n over two, times one plus n. That is formula number one. Right?

1:22Squares and cubes

Next, the squares. The small two up top means squared. One squared plus two squared, up to n squared, equals n times n plus one, times two n plus one, over six. Three factors on top, and six below. Write this down, it is formula number two. Clear? Now the cubes, with a small three up top. One cubed plus two cubed, up to n cubed. Remember our puzzle? So what do you guess this sum is? The square of the first sum! Correct! It is n times n plus one over two, and the big brackets with a small two outside mean the whole thing is squared. So the sum of the cubes is the square of the plain sum. Got it? Let's check with n equal to three. One plus eight plus twenty seven is thirty six, and six squared is thirty six. So our puzzle was no coincidence. This is formula number three, and it holds for every n. Okay?

2:21The trap: squares are not the square of the sum

Now, a quick question. The sum of the cubes was the square of the sum. So the sum of the squares must also be the square of the sum, right? No, no, careful! This is a classic trap. The equals sign with a slash through it means not equal. The sum of k squared is not the square of the sum of k. With me? Take n equal to three. One plus four plus nine is fourteen. But six squared is thirty six. Not even close! Right? So remember it this way: the square of the sum belongs to the cubes, never to the squares. Mix these up, and you lose easy marks. Clear?

2:59Polynomial terms: expand, then add

Now, the real power of these formulas. Say the k th term, T k, written as T with a small k below it, is a polynomial in k. Then expand it, and add term by term with our three formulas. That is the whole method! With me? Try one. One times two, plus two times three, plus three times four, and so on, up to n terms. The dot in the middle means times. The k th term is k times k plus one. How will you add it? Expand it! Very good! k times k plus one is k squared plus k. So add formula two and formula one, simplify, and you get n times n plus one times n plus two, over three. Beautiful, right? And for three factors in a row, k, k plus one and k plus two, the sum is n, n plus one, n plus two, n plus three, over four. See the pattern? Write one more factor on top, and divide by the number of factors you now have on top. Toppers write these two straight away. Okay? One last trap when you expand. Suppose a constant, say one, is left over. Its sum from k equals one to n is n, not one, because you add it n times. Many students forget this. Got it?

4:19Example 1: sum of squares and cubes from 1 to 20

Now, Example one, an easy warm-up. Find the sum of k squared, and the sum of k cubed, for k from one to twenty. That is sigma, with twenty on top. Pause the video, and try it first. Done? Okay, let's check. Here n is twenty. So, in the squares formula, what is two n plus one? Forty one. Perfect! So the sum of squares is twenty times twenty one times forty one, over six. Right? Now cancel before you multiply. Twenty over two is ten, and twenty one over three is seven. Ten times seven times forty one: two thousand eight hundred and seventy. Clear? Now the cubes. First the plain sum: twenty times twenty one, over two, is two hundred and ten. Then square it, okay? Two hundred and ten squared: forty four thousand one hundred. Done in two lines! Imagine adding twenty cubes by hand. No, thank you!

5:21Example 2: 9 terms of sum of cubes over sum of odd numbers

Now, a harder one, JEE style. Look at this series. The first term is one cubed over one. The second is one cubed plus two cubed, over one plus three. The third is one cubed plus two cubed plus three cubed, over one plus three plus five, and so on. Find the sum of the first nine terms. Pause the video and try it. Hint: find the n th term first. Done? Okay, let's check. In the n th term, the top is the sum of the first n cubes. And we know that one: n times n plus one over two, the whole squared. With me? Now the bottom. One. One plus three is four. One plus three plus five is nine. One, four, nine. What do you notice? Perfect squares! Yes, exactly! The bottom is the sum of the first n odd numbers, an A P with first term one and last term two n minus one. So it is n over two, times one plus two n minus one, which is n squared. Clear? Now divide. The n squared cancels, top and bottom. So T n, the n th term, is n plus one, the whole squared, over four. Look how simple that scary term became! Right? Now put n equal to one, two, up to nine. Then n plus one runs from two to ten. So S nine, written S with a small nine below it, the sum of nine terms, is one quarter of two squared plus three squared, up to ten squared. Okay? So, just use the squares formula with n equal to ten, right? Almost, but careful! The formula counts from the very first term, one squared. So first find one squared up to ten squared: ten times eleven times twenty one, over six. That is three hundred and eighty five. Got it? But our sum has no one squared in it, so subtract that one. Three hundred and eighty five minus one is three hundred and eighty four, and over four, that is ninety six. That is our answer. Always check where your sum starts, okay?

7:35Quick revision: sum of k, k^2, k^3 and the traps

So, quick revision. Three formulas, for the sums of k, k squared and k cubed, and the cubes give the square of the first one. For a polynomial term, expand, then add term by term. The sum of squares is not the square of the sum; that belongs to the cubes. The formulas start at k equals one, so always check where your sum starts. And a constant c sums to n times c. Now go and try the practice problems on this page. All the best!