How do I solve the recurrence relation f

Anahi Jensen

Anahi Jensen

Answered question

2022-05-23

How do I solve the recurrence relation f ( n ) = f ( n 1 ) + 3 n + 2

Answer & Explanation

exhumatql

exhumatql

Beginner2022-05-24Added 12 answers

f ( k ) = f ( k 1 ) + 3 k + 2
So
f ( k ) f ( k 1 ) = 3 k + 2
Adding both sides from k = 1 to n
f ( n ) f ( 0 ) = k = 1 n 3 k + 2
Which simpifies to
f ( n ) 2 = 3 2 n ( n + 1 ) + 2 n = n ( 3 n + 7 ) 2
Hence
f ( n ) = 2 + n ( 3 n + 7 ) 2 = 1 2 ( 3 n 2 + 7 n + 4 )
We can check several values
f ( 0 ) = 2
f ( 1 ) = 7
f ( 2 ) = 15

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