Geometric progression, how to find $x$

Question

I tried many ways to solve this problem, but I can't! Please, someone explain how to solve this problem:

If the sequence: $8, x, 50$ is a geometric progression, then $x = ?$

Answer 1

The sequence: $8, x, 50$ is a geometric progression then the ratio of the corresponding consecutive terms (i.e. the ratio of a term to its previous one) is constant. hence, we have $$\frac{x}{8}=\frac{50}{x}$$$$\implies x^2=400$$ $$\implies \color {#0b4} {x=\pm 20}$$ Note: As there is no information about the unknown term $x$ (i.e. either positive or negative) hence both the above values are acceptable satisfying the given condition (of a geometric progression) in the question.

Answer 2

You have that $$8:x = x:50$$ so $x^2=8 \cdot 50 = 400$, i.e. $x=20$.

Answer 3

Hint: What does it mean to be geometric? It means you get the next term by multiplying the previous by a common ratio, $r$. Then $x=8r$ and $50=xr$... can you solve these two equations for $x$?

Answer 4

For example, let $b_{0}, b_{1}, \ldots, b_{n}, \ldots$ - geometric progression. It means that $b_{k}=q^{k}b_{1}$. So, it's possible to derive that $b^{2}_{n}={b_{n-1} \cdot b_{n+1}}$