The sum, $S_n$, of the first $n$ terms of an arithmetic series is given by $S_n=3n^2+7n$. Find the first term and the common difference of the series.
$a = 10$ and $d = 6$
The half life of a radioactive isotope is the time it takes for a given mass to reduce to half the size of that given mass. A radioactive substance reduces from 12g to exactly 10.24g in 30 days. Assuming that the isotope decays exponentially, determine the half life of the isotope, correct to the nearest day.
$131$
The point $P$, with $y$ coordinate 2, lies on the curve $y=\dfrac{k+8k\sqrt{x}}{12x}$, where $k$ is a non zero constant. The tangent to the curve at $P$ is parallel to the line with equation $6x+y=17$. Find the value of $k$.
$24.7$