00:01
So according to the question here first of all the given sequence given sequence is something like this given sequence so it has the terms of 1 comma 1 then 3 then 9 then 21 then 41 and it is going and so on and we have to determine the nth term of this sequence okay, so first of all if we determine the first difference first difference for this sequence so 1 minus 1 will be 0 then 3 minus 1 it is 2 9 minus 3 is 6 21 minus 9 it is 12.
00:41
Okay, and 41 minus 21 it is 20 and it will go and so on now if we write the second difference second difference so 2 minus 0 is 2 6 minus 2 is 4 then 12 minus 6 it is 6 okay, 20 minus 12 it is 8 and it will go and so on so we can see that this is an arithmetic progression okay, so if we write the third difference third difference so it will be 4 minus 2 which is 2 6 minus 4 is 2 8 minus 6 is 2 so it will go and so on.
01:24
So this is a constant sequence that means constant term is repeating again and again, which is 2 2 2 okay, so we can say that formula for a n formula for a n because the third difference is repeating so we can say that formula for a n will be a 3 degree polynomial a 3 degree polynomial okay so a n that is our nth term it can be written as a basic 3 degree polynomial is a n cube plus b n square plus c n plus d.
02:06
Let this be our equation first okay now according to the question value of n is greater than or equal to 0 that means we have to consider n is equal to 0 also, okay so if we write the a 0 value that means the 0 term it is 1 then the first term a 1 term it is 1 according to the sequence we are writing a 2 term is 3 okay, and a 3 term it is 9 okay so first of all if you put n is equal to 0 then a 0 term will be these all n terms will become 0 so 0 plus 0 plus 0 and plus we will have d that means the value of d will be a 0 value is 1 so d value will be equal to 1 okay now we will put n is equal to 1 in the equation 1 so what we will get from here a 1 value will be a plus b plus c plus d value is 1 so a 1 value is 1 okay so 1 is equal to a plus b plus c plus 1 so 1 will be cancelled out from both the sides that means a plus b plus c.
03:06
It will be equal to 0 so let this be your equation 2.
03:10
Okay.
03:10
Now, let us put n is equal to 2 in equation 1 okay so we will get a 2 is equal to if we put 2 here so 2 q will be 8 so 8 a plus 4 b plus 2 c plus d value is 1 now a 2 value is 3 so 3 minus 1 will be 2.
03:29
Okay, so we can write it like this 8 a plus 4 b plus 2 c is equal to 2 okay, so what we can do so we have done a mistake here or what? like the n is equal to 2, okay, so what we have done from here so it is totally right because the n is equal to 2 value.
03:52
So 2 q is 8 so 8 a plus 4 b plus 2 c...