A geometric sequence has $a_{2} = 12$ and $a_{5} = 324$. What is the c…

Question

A geometric sequence has $a_{2} = 12$ and $a_{5} = 324$. What is the common ratio?

Accepted Answer

D. $3$

Use the relation $\dfrac{a_{5}}{a_{2}} = r^{5 - 2} = r^{3}$.
$\dfrac{324}{12} = 27 = r^{3} \Rightarrow r = \sqrt[3]{27} = 3$.
Verify: $a_{2} = 12$, $a_{3} = 36$, $a_{4} = 108$, $a_{5} = 324$ ✓.
- Trap A ($-3$) would give $a_{5} = 12 \cdot (-3)^{3} = -324$, sign wrong.
- Trap C ($4$) gives $a_{5} = 12 \cdot 64 = 768$ — too large.
- Trap D ($27$) is $r^{3}$, not $r$.
General relation: between any two terms of a geometric sequence, $\dfrac{a_{j}}{a_{i}} = r^{j - i}$.

Answer

$27$

Answer

$-3$

Answer

$4$