Question

the first three terms of a geometric sequence are as follows. 4, -8, 16 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, $r=\frac{-8}{4}=- 2$.

Step2: Find the fourth term

The $n$th term of a geometric sequence is given by $a_n=a_1r^{n - 1}$. For the fourth term ($n = 4$), $a_4=a_1r^{4 - 1}$, where $a_1 = 4$ and $r=-2$. So, $a_4=4\times(-2)^{3}=4\times(-8)=-32$.

Step3: Find the fifth term

For the fifth term ($n = 5$), $a_5=a_1r^{5 - 1}$, with $a_1 = 4$ and $r=-2$. So, $a_5=4\times(-2)^{4}=4\times16 = 64$.

Answer:

-32, 64