Find the nth term of the sequence (Explicit formula)

[AKJ1]

Homework Statement

{ 7/6 , 45/54 , 275/648, 1625/9720 ... }

Problem from a practice exam. My instructions are to find an explicit formula for the above sequence..

The Attempt at a Solution

Let A_n =7/6 , 45/54 , 275/648, 1625/9720 ...

Allow A_n = B_n / C_n

B_n = 7, 45, 275, 1625... (2n+5) 5^n-1 starting at n = 1

C_n = 6,54,648,9720... 3(n+1) [ What multiplies here is the preceding term, so if we allow n = 2 , we multiply the expression by 6 to get 54, if we allow n = 3, we multiply by 54 to get 648, if we allow n = 4, we multiply by 648 to get 9720.

I have no idea how to express this, or if there is a better way.

 

[cpsinkule]

C_n=C_n-13(n+1) and declare that C₀=1

 

[AKJ1]

C_n=C_n-13(n+1) and declare that C₀=1

∴ Starting at n=1 A_n = (2n+5) 5^n-1 / 3(n+1) C_n-1 , where we have declared C₀ = 1

Noting that B_n & C_n equal what we have defined above.

Correct?

 

[cpsinkule]

The entire sequence can't be indexed by two letters, A and C, you need to find a form where you only need A

 

[AKJ1]

The entire sequence can't be indexed by two letters, A and C, you need to find a form where you only need A

So I have to rewrite C_n in terms of A only? Is this possible? I am drawing a complete blank

 

[AKJ1]

Figured it out!