Find general term for a <mrow class="MJX-TeXAtom-ORD"> 2 n </mrow> </m

Finley Mckinney

Finley Mckinney

Answered question

2022-06-23

Find general term for a 2 n = ( 2 n 1 ) ( 2 n 2 ) a 2 n 2 + 2 ( 1 ) n 1 ( 2 n 2 ) !

Answer & Explanation

lilao8x

lilao8x

Beginner2022-06-24Added 22 answers

Let's solve
a 2 n = ( 2 n 1 ) ( 2 n 2 ) a 2 ( n 1 ) + 2 ( 1 ) n 1 ( 2 n 2 ) !
with a 2 = y ( 2 ) ( 0 ) = 2
Define
b n = ( 1 ) n + 1 a 2 n 2 ( 2 n 1 ) !
so
a 2 n = ( 1 ) n + 1 2 ( 2 n 1 ) ! b n
Substituting in the recurrence relation for a we have
( 1 ) n + 1 2 ( 2 n 1 ) ! b n = ( 2 n 1 ) ( 2 n 2 ) ( 1 ) n 2 ( 2 n 3 ) ! b n 1 + 2 ( 1 ) n 1 ( 2 n 2 ) !
Dividing by ( 1 ) n + 1 2 ( 2 n 1 ) ! we get
b n = b n 1 + 1 2 n 1
This, together with b 1 = 1 gives
b n = k = 1 n 1 2 k 1
Notice that this sum doesn't have a closed form.
So we end with
a 2 n = ( 1 ) n + 1 2 ( 2 n 1 ) ! k = 1 n 1 2 k 1

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