Question

If 4th term of a G.P. is 32 whose common ratio is half of the first term, then the 15th term is

• A. \(2^{12}\)
• B. \(2^{18}\)
• C. \(2^{14}\)
• D. \(2^{16}\)
• E. \(2^{10}\)

Hint:

Substitute relations between \(a\) and \(r\) early to reduce unknowns.

Answer 1

The Correct Option is D

Concept: Use \(T_n = ar^{n-1}\) and given relation between \(a\) and \(r\).

Step 1: Given
\[ T_4 = ar^3 = 32 \] Also: \[ r = \frac{a}{2} \]

Step 2: Substitute
\[ a\left(\frac{a}{2}\right)^3 = 32 \] \[ a \cdot \frac{a^3}{8} = 32 \] \[ \frac{a^4}{8} = 32 \] \[ a^4 = 256 \] \[ a = 4 \]

Step 3: Find \(r\)
\[ r = \frac{4}{2} = 2 \]

Step 4: Find 15th term
\[ T_{15} = ar^{14} = 4 \cdot 2^{14} \] \[ = 2^2 \cdot 2^{14} = 2^{16} \]

Step 5: Final answer
\[ \boxed{2^{16}} \]