If an AP has $S_{10}=3S_5$, what's the ratio of $u_{10}$ to $u_5$?

If an AP has $S_{10}=3S_5$, what's the ratio of $u_{10}$ to $u_5$?

Question

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 10th term to the 5th term.

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

Suggestion

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.

If $S_n$ is the sum of the first n terms, do we have a formula for $S_n$?

If $u_n$ is the nth term, do we have a formula for $u_n$?

Can we find the common difference with this added information?

Solution

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