Find a explicit formula for the recursion a_1=3, a_n=5a_{n-1}+12

Quessyrutty6w

Quessyrutty6w

Answered question

2022-07-15

Find a explicit formula for the recursion
a 1 = 3 a n = 5 a n 1 + 12
So i found
a 1 = 3 a 2 = 27 a 3 = 147 a 4 = 747

Answer & Explanation

Monica Dennis

Monica Dennis

Beginner2022-07-16Added 13 answers

Step 1
Try for characteristic equation:
a n = 5 a n 1 + 12 r = 5 a n = c 1 r n + c 2 a n = c 1 ( 5 ) n + c 2
Step 2
apply now a 1 = 3 , a 2 = 27 to find c 1 , c 2
a 1 = 3 = c 1 5 + c 2 a 2 = 27 = c 1 25 + c 2 a n = 6 5 ( 5 ) n 3 a n = 6 ( 5 ) n 1 3
owsicag7

owsicag7

Beginner2022-07-17Added 2 answers

Step 1
If substitute a n 2 , a n 3 , and so on, we have:
a n = 5 k a n k + 12 i = 0 k 1 5 i
a n = 5 k a n k + 12 5 k 1 5 1
a n = 5 k a n k + 3 ( 5 k 1 )
Step 2
Let: k = n 1, then n k = 1
a n = 5 n 1 a 1 + 3 ( 5 n 1 1 )

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