Question

In a geometric progression, the \(3^{rd}\) term is \(36\) and the \(5^{th}\) term is \(324\). The \(7^{th}\) term of the same progression will be _ _ _. (in integer)

Hint:

In a GP, dividing two terms eliminates the first term and helps directly determine the common ratio.

Answer 1

Correct Answer: 2916

Step 1: Recall the GP term.
The $n^{th}$ term is $T_n = a r^{n-1}$ with first term $a$ and ratio $r$.

Step 2: Use the two given terms.
$T_3 = a r^2 = 36$ and $T_5 = a r^4 = 324$.

Step 3: Find $r^2$.
Dividing, $\dfrac{a r^4}{a r^2} = r^2 = \dfrac{324}{36} = 9$.

Step 4: Step up to the 7th term.
Each two step jump multiplies by $r^2 = 9$. So $T_7 = T_5 \times r^2 = 324 \times 9 = 2916$.

Step 5: Answer.
\[ \boxed{2916} \]