Answer
$0.4286, 0.4615, 0.4737, 0.4800, 0.4839, 0.4865, 0.4884, 0.4898, 0.4909, 0.4918$
$\lim\limits_{n \to \infty}a_n=0.5$
Work Step by Step
It is given a sequence formulated by $a_n=\frac{3n}{1+6n}$
Substituting $n=1,2,\ldots,10$, we get the first ten terms of the sequence:
$0.4286, 0.4615, 0.4737, 0.4800, 0.4839, 0.4865, 0.4884, 0.4898, 0.4909, 0.4918$
Using these terms, the sequence appears to have a limit.
Calculate the limit:
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\frac{3n}{1+6n}$
$=\lim\limits_{n \to \infty}\frac{3}{\frac{1}{n}+6}$
$=\frac{3}{0+6}$
$=0.5$
So, the sequence converges to $0.5$
