Question

the first three terms of a geometric sequence are as follows. -243, -81, -27. find the next two terms of this sequence. give exact values (not decimal approximations).

Explanation:

Step1: Find the common ratio

The common ratio $r$ of a geometric sequence is found by dividing a term by its previous term. So, $r=\frac{-81}{-243}=\frac{1}{3}$.

Step2: Find the fourth term

The formula for the $n$th term of a geometric sequence is $a_n=a_1r^{n - 1}$. For the fourth term ($n = 4$), $a_4=a_3\times r$. Since $a_3=-27$ and $r=\frac{1}{3}$, then $a_4=-27\times\frac{1}{3}=-9$.

Step3: Find the fifth term

For the fifth term ($n = 5$), $a_5=a_4\times r$. Since $a_4=-9$ and $r=\frac{1}{3}$, then $a_5=-9\times\frac{1}{3}=-3$.

Answer:

-9, -3