An arithmetic progression is such that the sum of the first \(10\) terms is \(3\) times the sum of the first \(5\) terms. Find the ratio of the \(10\)th term to the \(5\)th term.

We know that for an arithmetic progression, if \(S_n\) is the sum of the first \(n\) terms, then

\[S_n =\dfrac{n}{2}(2a + (n-1)d).\]

Here we have that \[S_{10}=3S_5 \implies \dfrac{10}{2}(2a + 9d)=3\dfrac{5}{2}(2a + 4d) \implies 3d = a.\]

We also know that if \(u_n\) is the \(n\)th term, then \(u_n = a + (n-1)d.\)

Thus \[\dfrac{u_{10}}{u_5} = \dfrac{a+9d}{a+4d} = \dfrac{12d}{7d} = \dfrac{12}{7}.\]

Given further that the \(5\)th term is \(0.14\), calculate the sum of the first \(200\) terms.

If we have that \(a + 4d = 0.14\), then \(7d = 0.14\), and \(d = 0.02, a = 0.06\).

Now \[S_{200} = \dfrac{200}{2}(0.12 + (199)0.02)= 410.\]