Respuesta :
Step 1 - Write out the formula for the n-th term of a geometric sequence/progression.
[tex]\begin{gathered} T_n=ar^{n-1} \\ \text{where, a is the first term of the sequence, and r is the common ratio.} \end{gathered}[/tex]from the sequence question, a is 1/27
[tex]\begin{gathered} To\text{ find the common ratio, r,} \\ Divide\text{ any two consecutive terms.} \\ \text{for instance, } \\ \frac{T_2}{T_1}=\frac{\text{Second term}}{\text{First term}}\text{ }=\text{ }\frac{\frac{1}{9}}{\frac{1}{27}}\text{ }=\frac{1}{9}\text{ x }\frac{27}{1}\text{ }=3 \end{gathered}[/tex]Therefore the common ratio is r is 3 and the first term a is 1/27
We need both the first term and the common ratio to get the term that gives us a value of 6,561.
Step2 - substitute the values for the first term and common ratio into the n-th term formula earlier written.
[tex]\begin{gathered} 6561=\text{ }\frac{1}{27}X3^{n-1} \\ 6561\text{ X 27 }=3^{n-1} \\ 177147\text{ }=3^{n-1} \\ 3^{11}\text{ }=3^{n-1} \\ \text{Comparing the powers on both sides of the equation} \\ 11\text{ }=\text{ n -1} \\ n\text{ }=\text{ 11}+1 \\ n\text{ }=\text{ 12.} \end{gathered}[/tex]The term that gives 6561 in the sequence is 12
