## Task B-4.2.

Let $S_{n}, n \in \mathbb{N}$ be the sum of the first $n$ terms of a geometric sequence. If $S_{100}=10$ and $S_{300}=210$, what is $S_{400}$?
## Solution.

The sum of the first $n$ terms of a geometric sequence is calculated by the formula $S_{n}=a_{1} \frac{q^{n}-1}{q-1}, q \neq 1$. From the given sums in the problem, we see that the sequence is not constant, i.e., $q \neq 1$.

Then we have $a_{1} \frac{q^{100}-1}{q-1}=10$ and $a_{1} \frac{q^{300}-1}{q-1}=210$.

If we divide the second equation by the first, we get $\frac{q^{300}-1}{q^{100}-1}=21$.

The numerator of this fraction can be factored as a difference of cubes, and then we have:

$$
\begin{gathered}
\frac{\left(q^{100}\right)^{3}-1}{q^{100}-1}=21 \\
\frac{\left(q^{100}-1\right)\left(q^{200}+q^{100}+1\right)}{q^{100}-1}=21 \\
q^{200}+q^{100}-20=0
\end{gathered}
$$

If we let $x=q^{100}$, we get the quadratic equation $x^{2}+x-20=0$ whose solutions are $x=-5$ and $x=4$.

Thus, $q^{100}=-5$ and $q^{100}=4$. An even power of a real number cannot be negative, so the only solution is $q^{100}=4$.

It remains to determine the first term of the sequence. Express the first term $a_{1}$ from $a_{1} \frac{q^{100}-1}{q-1}=10$.

$a_{1}=\frac{10(q-1)}{q^{100}-1}=\frac{10\left(4^{\frac{1}{100}}-1\right)}{4-1}=\frac{10}{3}\left(4^{\frac{1}{100}}-1\right)$.

Then the desired sum is

$$
S_{400}=a_{1} \frac{\left(q^{100}\right)^{4}-1}{q-1}=\frac{10}{3}\left(4^{\frac{1}{100}}-1\right) \frac{4^{4}-1}{\left(4^{\frac{1}{100}}-1\right)}=850
$$

## Note.

If the difference of cubes is not used to solve the obtained system of equations, the system can be reduced to the equation $x^{3}-21 x+20=0, x=q^{100}$. This equation can be solved by factoring or by considering the divisors of the constant term. Factoring:

$$
\begin{aligned}
x^{3}-21 x+20 & =x^{3}-x^{2}+x^{2}-20 x-x+20= \\
& =x^{2}(x-1)+x(x-1)-20(x+1)= \\
& =(x-1)\left(x^{2}+x-20\right)= \\
& =(x-1)(x-4)(x+5)
\end{aligned}
$$

Since $q \neq 1$, the only positive solution is 4. The rest of the process is as described in the solution.