How to find explicit formula for a sequence seems to be geometric but isn't? The problem asks for an explicit formula for the following recursive formula: a_1=2, a_n=5(a_{n-1}+2)

Jairo Hodges

Jairo Hodges

Answered question

2022-11-18

How to find explicit formula for a sequence seems to be geometric but isn't?
I recently came across this interesting problem, where the progression was neither arithmetic nor geometric. The problem asks for an explicit formula for the following recursive formula:
a 1 = 2
a n = 5 ( a n 1 + 2 )
So the first five terms are 2,20,110,560,2810.
I tried to distribute the 5 in the second equation to get 5 a n 1 + 10 and then tried to apply the formula for a geometric sequence, which is a 1 ( r ) n 1 , but got only 2 ( 5 ) n 1 , which clearly doesn't work for the sequence.
I also tried using finite differences on the first five terms to see if it was just a polynomial rule, but that didn't work.
How would you go about doing a problem like this?

Answer & Explanation

avuglantsaew

avuglantsaew

Beginner2022-11-19Added 15 answers

Step 1
First begin by solving, as to why should be evident latter,
L = 5 ( L + 2 )
The solution is L = 2.5, which we call a particular solution. Then we may write,
(1) 2.5 = 5 ( 2.5 + 2 )
Recall that the what we want to solve is,
(2) a n = 5 ( a n 1 + 2 )
Luckily for us by subtracting the first equation from the second we get,
a n + 2.5 = 5 ( a n 1 + 2.5 )
Step 2
Let b n = a n + 2.5. The recursion transforms to (a homogenous linear recurrence):
b n = 5 b n 1
Because we must multiply by five each step, the solution to this is clearly,
b n = 5 n 1 b 1
But by definition b 1 = a 1 + 2.5 and a n = b n 2.5. So we get,
a n = 5 n 1 ( a 1 + 2.5 ) 2.5

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