00:01
Hello everyone, in this question we have to find two things.
00:03
So first thing is we need to prove that 1 square plus 2 of square plus up to n of square this is equals to n times of n plus 1 times of 2 n plus 1 whole divided by 6 where n is greater than equals to 1.
00:20
Now we have to prove this.
00:22
So we can prove this by induction.
00:24
So let's assume first of all for n equal to 1 if we want to prove.
00:28
So for n is equals to 1 for left hand side what we have n is equals to 1 just put it there.
00:34
So we have that is 1 of square is equals to 1 and for right hand side when we put n is equals to 1 we have that is 1 times of 1 plus 1 that is 2 then 2 times of 1 plus 1 that is clearly 3 and this whole divided by 6.
00:48
So we can say that this is equals to 1 right.
00:52
So we have that left hand side is equals to right hand side.
00:54
So we can be so we can say that that the above expression is true for n equal to 1 right.
01:04
Now for n equal to 1 we have proved.
01:07
Now let's assume for n is equals to k this is true.
01:12
We are assuming this.
01:14
So this part is we assume that is 1 of square plus 2 of square plus k of square this is equals to we have k times of k plus 1 then 2 times of k plus 1 whole divided by 6.
01:31
Let's assume this this is true for all k right.
01:36
Now further to prove this what we have to do now we have to check is this is true for n is equals to k plus 1 or not.
01:47
So suppose if we take left hand side so we have left hand side that is 1 of square plus 2 of square plus up to if n is equals to k plus 1.
01:56
So we have that is k plus 1 of whole square right.
01:59
Now this can be written as 1 of square plus 2 of square plus clearly k plus 1 of square that means the term before k plus 1 is k right.
02:10
So we can write k square plus k plus 1 of whole square.
02:14
Now the value of this expression we have here right we can put it there.
02:19
So we can just write the above expression as k times of k plus 1 then 2 times of k plus 1 divided by 6 and plus the k plus 1 of whole square term as it is right.
02:32
Now further take common that is k plus 1.
02:35
So what we have that is k times of 2 times of k plus 1 divided by 6 plus k plus 1 right.
02:44
Now take lcm and we have 1 by 6 is common and k plus 1 as it is.
02:50
So we have k times of 2 k plus 1 and plus 6 times of k plus 1 right.
02:58
Now further solving this what we have that is 1 by 6 k plus 1 and when we take multiply the whole term so we have that is 2 k square plus k plus 6 of k plus 6 right.
03:15
So this implies 1 of by 6 k plus 1 times of that is now we have 2 k square plus k plus 6 k right.
03:26
So what we can just do that is we can write it this as 2 k square plus 7 k plus 6 right.
03:34
So this implies we have 1 by 6 times of k plus 1 and this can be written as 2 k square.
03:41
Now 7 k can be by middle terms of splitting what we get that is 4 k plus 3 k plus 6 right.
03:51
Now further what we have that is 1 by 6 k plus 1 and this implies when we take common that is 2 of k so we get k plus 2 right and plus when we take common 3 so we have again k plus 2 common.
04:09
So now we can take common as k plus 2 so 1 by 6 k plus 1 and then we can take common k plus 2 so what we left with 2 k plus 3.
04:19
So this is our this is our left hand side.
04:23
Now when we calculate for right hand side and just put n is equal to k plus 1.
04:30
So let's see what we get.
04:32
So clearly we have that is further n n plus 1 2 n plus 1 divided by 6 right.
04:40
So just we have to put the value of n there...