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 general term of a GP.
The \(n^{th}\) term of a geometric progression is given by
\[ T_n=ar^{n-1} \] where
\[ a=\text{first term}, \qquad r=\text{common ratio}. \]

Step 2: Write the given terms.
The \(3^{rd}\) term is
\[ T_3=ar^2=36 \] The \(5^{th}\) term is
\[ T_5=ar^4=324 \]

Step 3: Divide the equations.
\[ \frac{T_5}{T_3} = \frac{ar^4}{ar^2} = r^2 \] \[ r^2=\frac{324}{36} \] \[ r^2=9 \]

Step 4: Find the common ratio.
\[ r=\pm 3 \] For finding the \(7^{th}\) term, \(r^2=9\) is sufficient.

Step 5: Find the first term.
Using
\[ ar^2=36 \] and \(r^2=9\),
\[ a\times 9=36 \] \[ a=4 \]

Step 6: Find the \(7^{th}\) term.
\[ T_7=ar^6 \] Since
\[ r^2=9, \] \[ r^6=(r^2)^3=9^3=729 \] Thus,
\[ T_7=4\times 729 \] \[ T_7=2916 \]

Step 7: Final conclusion.
Therefore, the \(7^{th}\) term of the GP is
\[ \boxed{2916} \]