Respuesta :
We have the sequence 3, 6, 9, 12.
We can prove that this sequence is not geometric by calculating the ratios a2/a1 and compare it to a3/a2. If the ratios are not equal, the sequence is not geometric: there is no common ratio in the sequence.
[tex]\begin{gathered} \frac{a_2}{a_1}=\frac{6}{3}=2 \\ \frac{a_3}{a_2}=\frac{9}{6}=1.5 \\ \frac{a_2}{a_1}\neq\frac{a_3}{a_2}\Rightarrow\text{ not a geometric sequence} \end{gathered}[/tex]We can prove it is a arithmetic sequence because it has a common difference d=3 between consecutive terms:
[tex]\begin{gathered} a_2=a_1+3=3+3=6 \\ a_3=a_2+3=6+3=9 \end{gathered}[/tex]Now, we have to calculate the sum.
We start by calculating a10, using the explicit function for an arithmetic sequence:
[tex]a_{10}=f(10)=3+3\cdot(10-1)=3+3\cdot9=3+27=30[/tex]Then, we apply the formula given for the sum of n terms:
[tex]\begin{gathered} S=\frac{n}{2}(a_1+a_n) \\ S_{10}=\frac{10}{2}(3+30)=5\cdot33=165 \end{gathered}[/tex]Answer:
The sequence is arithmetic.
The sum of the first 10 terms is S10=165.
