This Question Looks Hard...But Actually It's Easy! | Newtons Sum

[00:00:02]Let's get it started is straight away by writing this is equation one and here we'll write equation two. Let's say required expression is E.

[00:00:13]I will simplify E first. So, we can write E will be X + Y + X cube over Y square + Y cube over X square.

[00:00:28]Now, in place of X + Y we will write four because of equation number one.

[00:00:37]But, here we have X cube over Y square + Y cube over X square. So, let's convert our denominators into common denominator. We need to multiply here with X square.

[00:00:48]So, we need to multiply here with Y square.

[00:00:52]Now, we'll be writing E equal to 4 + in numerator X cube * X square X power five Y cube * Y square Y power five over X square Y square, which we'll be writing XY whole square because we have equation two also.

[00:01:17]So, in place of XY we'll be writing minus two.

[00:01:21]So, 4 + X power five + Y power five over minus two whole square.

[00:01:31]Minus two whole square is plus four.

[00:01:34]We will get 4 + X power five + Y power five over four.

[00:01:43]This is our target expression. First, we will find X power five + Y power five.

[00:01:51]Then, we will be able to find E.

[00:01:55]I'll be using Newton's sum or recurrence formula method.

[00:02:01]So, let us think about one quadratic equation whose roots are x + y. Then here we have sum of roots.

[00:02:10]x * y = -2. This is product of roots.

[00:02:15]Let's write quadratic equation in variable t whose roots are x and y. Then we can write t squared minus sum of roots 4 t plus product of roots minus 2 equal to 0.

[00:02:35]We'll add 4t plus 2 to both sides and we can write t squared equal to 4t plus 2.

[00:02:46]Now we have to consider sum sn will be equal to x power n plus y power n.

[00:02:57]From our quadratic equation t squared equal to 4t plus 2 we can write sn equal to 4 times s n minus 1 plus 2 times s n minus 2.

[00:03:16]Holds true for n greater than or equal to 2. So, we have sn over here and we have sum sn over here.

[00:03:28]Now I will put n equal to 0.

[00:03:31]So, I will be able to get s0 which will be equal to x power 0 plus y power 0, 1 plus 1, 2.

[00:03:41]Now I will put n equal to 1.

[00:03:44]So, we will get s1 will be equal to x power 1 plus y power 1 which will be x plus y as per equation one, value is four.

[00:03:56]In the same way, if I will put n equal to five, we are going to get s five, which will be equal to x power five plus y power five.

[00:04:08]Now, we will write our required expression in terms of s five.

[00:04:14]It was four plus x power five plus y power five over four.

[00:04:22]So, required expression e can be written in terms of s five, four plus s five over four.

[00:04:33]So, we have to find s five from here.

[00:04:37]Now, I will put n equal to two over here.

[00:04:43]So, we will get s two equal to four s one plus two s zero.

[00:04:52]We have s one and s zero over here. s one is four, s zero is two.

[00:04:59]So, we will write four times four plus two times two.

[00:05:06]Will be equal to 16 plus four.

[00:05:10]So, s two is 20.

[00:05:13]Now, we have s zero two, s one four, s two 20. Let me write here.

[00:05:19]s zero two s one four s two 20.

[00:05:27]And we have general s n equal to four times s n minus one plus two times s n minus two.

[00:05:38]Now, we will plug in n equal to three.

[00:05:41]And we will get s three.

[00:05:44]Which will be four times s two plus two times s one.

[00:05:50]We have S2 20 and we have S1 4.

[00:05:56]So, we can easily calculate S3 from here. 4 * 20 + 2 * 4.

[00:06:05]4 * 20 is 80.

[00:06:08]2 * 4 is 8.

[00:06:11]So, we have S3 88.

[00:06:15]Now, we have to put n equal to 4.

[00:06:20]So, let me write here n equal to 4.

[00:06:24]Sn is 4 * Sn - 1 + 2 * Sn - 2.

[00:06:32]We will get S4 4 * S3 + 2 * S2.

[00:06:41]S2 is 20. Let me write here S0 is 2.

[00:06:46]S1 is 4.

[00:06:48]S2 is 20.

[00:06:51]S3 is 88.

[00:06:55]And we have to calculate S4. We will write here 4 * 88 + 2 * S2. It is 20.

[00:07:05]4 * 88 is 352 + 40.

[00:07:11]352 + 40 is 392.

[00:07:15]Now, we have S4 also.

[00:07:20]Now, we have to calculate S5, which is required as per our expression.

[00:07:26]Now, I will put over here n equal to 5 to get the value of S5, which will be equal to 4 * S4 + 2 * S3.

[00:07:37]S4 is 392. We will write 4 * 392 + 2 * S3. S3 is 88.

[00:07:48]4 * 392 is 1,568.

[00:07:54]Plus 2 * 88 is 176.

[00:07:58]We'll add both the numbers to get the value of S5. It is 1,744.

[00:08:08]S5 is known.

[00:08:09]Now, we will calculate required expression E.

[00:08:13]E is 4 + S5 over 4.

[00:08:21]And S5 is 1744.

[00:08:26]Let's put 4 + 1744 over 4.

[00:08:35]So, we'll write here 4 + 436.

[00:08:42]4 + 436 will be 440.

[00:08:47]So, our required expression would be equal to 440 using Newton's sums or recurrence relationship.

[00:08:57]I hope, friends, you will like this video. Thank you so very much for watching. Do not forget to like, share, and subscribe.

[00:09:04]Bye-bye.