00:02
In this question, we are asked to solve the recurrence relation.
00:05
We are given a .n is equal to 2 a .n subtracted by 1, negative of 3 times of 3 raise to the power of n.
00:15
And we are also given the initial condition which is equal to a0 equals to 2.
00:21
So now first let us say g of x is equal to summation n is equal to 0 to infinity.
00:30
A n x to the power of n.
00:33
So this is going to be the generating function for the sequence a .n.
00:43
So now we are given this condition.
00:46
So now we have to multiply each term in the recurrence relation.
00:50
We are going to multiply each terms of the recurrence relation by x to the power of n and summing from 1 to infinity.
01:01
So this is going to be x summation n is equals to 1 to infinity, a base to the core base n, x to the core of n is equals to 2 times of summation n is equals to 1 to infinity, a n negative 1, x to the power of n, subtracted by 3 times of summation n is equals to 1 to infinity, 3 raise to the power of n, x to the power of n.
01:27
So now let us say that g of x subtracted by a0, which is the given initial condition.
01:36
So this is going to be two times x multiplied by g of x.
01:41
So this is going to be subtracted by three times of we have summation expansion.
01:48
So summation n is equals to 0 to infinity.
01:51
3 raised to the power of n, x to the power of n, subtracted by 3 raised to the power of 0, x to the power of 0.
01:58
So now similarly, we have to expand this further.
02:04
So we'll be having 2 times of x, g of x negative of 3 times of 1 over 1 negative of 3x subtracted by 1.
02:16
So now this is actually because we know that recalling this x g of x equals to summation n is equal to 0 to infinity...