We can find the common ratio with the following formula:
[tex]r=\frac{a_{n+1}}{a_n}[/tex]
In this case, we have the following:
[tex]\begin{gathered} r_1=\frac{a_2}{a_1}=\frac{10}{5}=2 \\ r_2=\frac{a_3}{a_2}=\frac{20}{10}=2 \end{gathered}[/tex]
we can see that the common ratio is r = 2. Then, we have the following formula for the sequence:
[tex]\begin{gathered} a_n=a_1\cdot r^{n-1} \\ a_1=5 \\ r=2 \\ \Rightarrow a_n=5\cdot2^{n-1} \end{gathered}[/tex]
Now, to find the 10th term,we make n = 10:
[tex]\begin{gathered} n=10 \\ \Rightarrow a_{10}=5\cdot2^{10-1}=5\cdot2^9=5\cdot512=2560 \end{gathered}[/tex]
therefore, the 10th term is 2560
The sum of the series can be calculated with the formula:
[tex]\begin{gathered} S_n=\frac{5(1-2^n)}{1-2}=-5(1-2^n) \\ \end{gathered}[/tex]
