Question

Find the common ratio and the number of terms in the given finite geometric sequence.a_n={2,1, \frac{1}{2} ,..., \frac{1}{1024} }The common ratio is AnswerThere are Answer terms.

Answer 1

Notice that:

$\begin{gathered} 1=2*\frac{1}{2}, \\ \frac{1}{2}=1*\frac{1}{2}. \end{gathered}$

Therefore, the common ratio is:

$\frac{1}{2}.$

The mth term of the sequence has the general form:

$a_m=2(\frac{1}{2})^{m-1}.^$

Setting

$a_m=\frac{1}{1024},$

we get:

$\frac{1}{1024}=2(\frac{1}{2})^{m-1}.$

Solving the above equation for m, we get:

$\begin{gathered} \frac{1}{2*1024}=\frac{1}{2^{m-1}}, \\ 2048=2^{m-1}, \\ log_2(2048)=m-1(log_22), \\ 11=m-1, \\ m=11+1, \\ m=12. \end{gathered}$

Finally, we get that there are

$12$

terms in the finite geometric sequence.

Answer:

First blank:

$\frac{1}{2},$

Second blank:

$12.$