Given:
[tex]\begin{gathered} a_3=45 \\ a_6=1215 \end{gathered}[/tex]The formula for general term is:
[tex]a_n=a_1r^{n-1}[/tex]Thus, we can write:
[tex]\begin{gathered} a_3=a_1r^2=45 \\ a_6=a_1r^5=1215 \end{gathered}[/tex]We can divide a_6 by a_3:
[tex]\begin{gathered} \frac{a_6}{a_3} \\ \frac{a_1r^5}{a_1r^2}=\frac{1215}{45} \\ \frac{r^5}{r^2}=27 \\ r^{5-2}=27 \\ r^3=27 \\ r=\sqrt[3]{27} \\ r=3 \end{gathered}[/tex]We found r = 3.
Now, we need first term, a_1.
So,
[tex]\begin{gathered} a_3=a_1r^2=45 \\ a_1(3)^2=45 \\ 9a_1=45 \\ a_1=\frac{45}{9} \\ a_1=5 \end{gathered}[/tex]So,
First Term = 5
Common Ratio = 3
The general term of this geometric sequence is:
So,
First Term = 5
Common Ratio = 3
The general term of this geometric sequence is:
