Sequence and Series
Infinite Exponents
JEE Main 2020
Grade 11
Question:
The product $2^{1/4} \cdot 4^{1/16} \cdot 8^{1/48} \cdot 16^{1/128} \dots$ to $\infty$ equals $2^k$. Find $k$ (as a fraction in lowest terms, expressed as $p/q$; enter $p + q$).
Step-by-Step Solution
Key Concept: Write each factor as $2^{n/4^n}$ and check that the exponents $1/4, 1/8, 1/16, \dots$ form a G.P. Sum the infinite G.P. to get the total exponent.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 3