Question

Find the 12th term in the following geometric sequence. 0.75, 1.5, 3, 6, . . .

Answer 1

We know that this is a geometric sequence.  Keep in mind that in a geometric sequence, the next term is found by shamelessly multiplying it by "some number", namely the "common ratio".

therefore, if you divide any "next term" by the "previous term", the quotient will just be that "some number", thus, if 1.5 ÷ 7.5 will give it to us, and is, yeap, you guessed it, is 2.

you can check say 3 ÷ 1.5 or 6 ÷ 3, you'll see it.

so, we know the common ratio is 2, and the first term is 0.75, so

 $\bf n^{th}\textit{ term of a geometric sequence}\\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ r=2\\ a_1=0.75\\ n=12 \end{cases} \\\\\\ a_{12}=0.75\cdot 2^{12-1}\implies a_{12}=0.75\cdot 2^{11}\implies a_{12}=1536$