+14 1) Given that \( a_0 = 1\) and \( a_1 = 5 \), and the general relation \(a_n^2 - a_{n - 1} a_{n + 1} = (-1)^n\)
for \( n \ge 1,\) find \(a_{10}.\)
2)Evaluate the sum. \($\dfrac{6}{3^2-1}+\dfrac{6}{5^2-1}+\dfrac{6}{7^2-1}+\dfrac{6}{9^2-1}+\cdots$.\)
3)\(A sequence $\{a_n\}$ satisfies $a_1 = 1$ and \[a_n = \frac{a_{n - 1}}{1 + a_{n - 1}}\]for all $n \ge 2.$ Find $a_{10}.$ \)
4)The sum of the first n terms of a certain sequence is n(n+1)(n+2) Find the tenth term of the sequence.
+9673 1.
(Wow, this is some really advanced homework)
The generated sequence is 1,5,26,135,701,..., which is OEIS A052918 (Took me a while to find it). And a10 is 2,646,275.
Notes: The general relation can be rewritten as an+1 = 5an + an-1.
https://oeis.org/A052918
^ Here's a link to the sequence.
+9673 2.
\(\dfrac{6}{3^2-1} + \dfrac{6}{5^2-1} +...\\ = \displaystyle 6\sum^{\infty}_{n=0} \dfrac{1}{(2n+3)^2 - 1}\\ = \displaystyle \dfrac{3}{2}\sum^{\infty}_{n=0} \dfrac{1}{(n+1)(n+2)}\\ = \displaystyle \dfrac{3}{2}\sum^{\infty}_{n=0} \left(\dfrac{1}{n+1} - \dfrac{1}{n+2}\right)\\ = \displaystyle \dfrac{3}{2}\left(\sum^{\infty}_{n=0} \dfrac{1}{n+1} - \sum^{\infty}_{n=0}\dfrac{1}{n+2}\right)\\ = \displaystyle \dfrac{3}{2}\left(\sum^{\infty}_{n=0} \dfrac{1}{n+1} - \sum^{\infty}_{n=1}\dfrac{1}{n+1}\right)\\ = \dfrac{3}{2}\left(\dfrac{1}{0+1}\right)\\ =\dfrac{3}{2} \)
