Question

The 5th and 7th terms of a G.P. are 12 and 48 respectively. Then the $9^{\text{th}}$ term is

• A. 162
• B. 96
• C. 192
• D. 144
• E. 182

Hint:

Shortcut Tip: Since the term positions (5, 7, 9) form an arithmetic sequence, the term values themselves (12, 48, $x$) form a geometric sequence! The multiplier is $48/12 = 4$, so $x = 48 \times 4 = 192$.

Answer 1

The Correct Option is C

Concept:
The $n$-th term of a Geometric Progression (G.P.) is given by $a_n = a \cdot r^{n-1}$, where $a$ is the first term and $r$ is the common ratio. In any G.P., the ratio between terms separated by $k$ steps is exactly $r^k$.

Step 1: Write equations for the given terms.
Using the general formula $a_n = a \cdot r^{n-1}$: $$a_5 = a \cdot r^4 = 12$$ $$a_7 = a \cdot r^6 = 48$$

Step 2: Determine the ratio multiplier ($r^2$).
Divide the 7th term equation by the 5th term equation to isolate the ratio: $$\frac{a_7}{a_5} = \frac{a \cdot r^6}{a \cdot r^4}$$ $$\frac{48}{12} = r^2$$ $$r^2 = 4$$

Step 3: Express the 9th term using knowns.
We need to find the 9th term ($a_9$). We can express it relative to the 7th term rather than starting from scratch: $$a_9 = a \cdot r^8$$ $$a_9 = (a \cdot r^6) \cdot r^2$$ $$a_9 = a_7 \cdot r^2$$

Step 4: Substitute values into the expression.
Substitute the known values $a_7 = 48$ and $r^2 = 4$ into our expression: $$a_9 = 48 \times 4$$

Step 5: Calculate the final answer.
Multiply the numbers to get the 9th term: $$a_9 = 192$$ Hence the correct answer is (C) 192.