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is it correct way to solve this?
if not then please provide me the solution

No, it is wrong. It is non homogeneous recurrence relation.

I have done the simplification,now you just need to put the value of aa1  and get the answer

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welcome always!
how do you get coefficient of 1st term as 2 after putting reasonable solution?

as the rhs of equation 2.1n  so we tool a trial of 2A1n

and put this value to original equation

i made one silly misrtake! now updated