Test Index

Sequence and Series

Question : 68 of 86

Marks: +1, -0

The value of

A^{\frac{1}{2}} \cdot A^{\frac{1}{4}}, A^{\frac{1}{8}}

to infinity is -

Solution:

Concept:

The product of powers with the same base is the base raised to the sum of the exponents.

Explanation:

Step 1: Write the product as

A^{\frac{1}{2}} \cdot A^{\frac{1}{4}} \cdot A^{\frac{1}{8}} \cdots

Step 2: Using

a^m \cdot a^n = a^{m+n}

, the product equals

A^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots}

Step 3: The exponents form an infinite geometric series: first term

a = \frac{1}{2}

, common ratio

r = \frac{1}{2}

Step 4: Sum of the series

S = \frac{a}{1-r} = \frac{1/2}{1-1/2} = 1

Step 5: Therefore, the product is

A^1 = A

Answer:

A

Go to Question:

Prev Question

Next Question