0
$\begingroup$

I need to find an explicit definition for the recursive sequence $$a_0=2,\space a_n=a_{n-1}*(2+4n)$$ My first instinct was to do something like $a_n=2(2+4n)^n$. However, that doesn't work because the common ratio isn't $2+4n$ where n is treated as a constant, it changes for each term, meaning the sequence looks like this: $$2*(2+4)*(2+8)*(2+12)*(2+16)*\space...$$ Any ideas?

$\endgroup$
1
$\begingroup$

$$ \prod_{n=1}^j\frac{a_{n}}{a_{n-1}}=\prod_{n=1}^j(2+4n)=2^j\prod_{n=1}^j(2n+1)=2^j(2j+1)!!\\ \implies\frac{a_j}{a_0}=2^j(2j+1)!!\\ \implies a_j=2\frac{(2j+1)!}{j!} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.