[tex]\begin{gathered} \text{ If we like to see if the sequence is arithmetic, the difference of two consecutive terms} \\ \text{ must remain constant, but} \\ 10-2=8 \\ 50-10=40 \end{gathered}[/tex][tex]\begin{gathered} \text{ So we suspect the sequence is geometric, in that case, the division of two consecutive} \\ \text{ terms must be constant} \\ \frac{10}{2}=5 \\ \frac{50}{10}=5 \\ \frac{250}{50}=5 \\ \text{ Thus we indeed have a geometric sequence!} \end{gathered}[/tex][tex]\begin{gathered} \text{and we find that the subsequent terms are obtained multiplying the previous one by 5 } \\ \text{ So the next five terms are:} \\ 250\cdot5=1250 \\ 1250\cdot5=6250 \\ 6250\cdot5=31250 \\ 31250\cdot5=156250 \\ 156250\cdot5=781250 \end{gathered}[/tex][tex]\begin{gathered} \text{ Now, we find an explicit formula for the sequence by} \\ 10=2\cdot5 \\ 50=2\cdot5^2 \\ 250=2\cdot5^3 \\ So\text{ we have } \\ a_n=2\cdot5^n \end{gathered}[/tex][tex]\begin{gathered} \text{ Since the next term is obtained by multiplying the previous one by 5, we have} \\ a_0=2, \\ a_{n+1}=5a_n\text{ for n}\ge0 \end{gathered}[/tex]
