Key Concepts and Formulas

1. Properties of Exponents: The fundamental property used is $a^m \cdot a^n = a^{m+n}$, which allows us to convert a product of powers with the same base into a single power whose exponent is the sum of the individual exponents.
2. Sum of an Infinite Geometric Progression (GP): For a geometric progression with first term $a$ and common ratio $r$, where $|r| < 1$, the sum to infinity is given by $S_{\infty} = \frac{a}{1-r}$.
3. Sum of an Infinite Arithmetico-Geometric Progression (AGP): A series of the form $a, (a+d)r, (a+2d)r^2, \dots$ is an AGP. The sum of such a series can be found by multiplying the series by its common ratio $r$ (where $|r|<1$), shifting the terms, and subtracting the new series from the original one, which results in a GP.

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Step-by-Step Solution

We are asked to find the value of the infinite product: $P = {2^{1/4}} \cdot {4^{1/8}} \cdot {8^{1/16}} \cdot \dots \infty$

Step 1: Express all terms with a common base. To combine the terms using exponent properties, we need a common base. The smallest common base is $2$. The terms can be rewritten as:

• $2^{1/4}$
• $4^{1/8} = (2^2)^{1/8} = 2^{2/8}$
• $8^{1/16} = (2^3)^{1/16} = 2^{3/16}$
• The general term appears to be $(2^n)^{1/2^{n+1}} = 2^{n/2^{n+1}}$.

Thus, the product becomes: $P = 2^{1/4} \cdot 2^{2/8} \cdot 2^{3/16} \cdot 2^{4/32} \cdot \dots \infty$

Step 2: Convert the product into a sum in the exponent. Using the property $a^m \cdot a^n \cdot a^p \dots = a^{m+n+p+\dots}$, we can write the product $P$ as a single base raised to the sum of the exponents: $P = 2^{\left(\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \dots \infty\right)}$ Let $S$ be the sum of the exponents: $S = \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \dots \infty \quad \dots (1)$

Step 3: Calculate the sum of the infinite series $S$. The series $S$ is an Arithmetico-Geometric Progression (AGP). The numerators form an AP ($1, 2, 3, 4, \dots$) with first term $a_A=1$ and common difference $d=1$. The denominators form a GP ($4, 8, 16, 32, \dots$) which can be seen as $4 \cdot (2^0), 4 \cdot (2^1), 4 \cdot (2^2), 4 \cdot (2^3), \dots$. Alternatively, we can write the terms of $S$ as: $S = \frac{1}{4} + \frac{2}{4 \cdot 2} + \frac{3}{4 \cdot 2^2} + \frac{4}{4 \cdot 2^3} + \dots$ The common ratio of the geometric part of this AGP is $r = \frac{1}{2}$.

To find the sum of this infinite AGP, we multiply the series by $r = 1/2$ and subtract: Multiply equation (1) by $1/2$: $\frac{1}{2}S = \frac{1}{2} \left( \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \dots \right)$ $\frac{1}{2}S = \quad \frac{1}{8} + \frac{2}{16} + \frac{3}{32} + \dots \infty \quad \dots (2)$

Now, subtract equation (2) from equation (1): $S - \frac{1}{2}S = \left( \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \dots \right) - \left( \frac{1}{8} + \frac{2}{16} + \frac{3}{32} + \dots \right)$ Aligning terms with the same denominator for subtraction: $\frac{1}{2}S = \frac{1}{4} + \left( \frac{2}{8} - \frac{1}{8} \right) + \left( \frac{3}{16} - \frac{2}{16} \right) + \left( \frac{4}{32} - \frac{3}{32} \right) + \dots$ $\frac{1}{2}S = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \dots \infty$

The series on the right-hand side is now an infinite Geometric Progression (GP) with:

• First term, $a = \frac{1}{4}$
• Common ratio, $r = \frac{1/8}{1/4} = \frac{1}{2}$

Since $|r| = |1/2| < 1$, the sum to infinity of this GP exists and is given by $S_{\infty} = \frac{a}{1-r}$. $S_{\infty} = \frac{1/4}{1 - 1/2} = \frac{1/4}{1/2} = \frac{1}{4} \times 2 = \frac{1}{2}$

So, we have: $\frac{1}{2}S = \frac{1}{2}$ Multiplying both sides by $2$ gives: $S = 1$

Step 4: Substitute the sum back into the expression for $P$. We found that the exponent $S = 1$. Substitute this value back into the expression for $P$: $P = 2^S = 2^1 = 2$

The value of the given product is $2$.

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Common Mistakes & Tips

• Identify the AGP correctly: The series in the exponent must be recognized as an AGP. The terms are of the form $\frac{n}{2^{n+1}}$, which fits the pattern of an AGP.
• Careful subtraction of series: When calculating the sum of the AGP, ensure that terms are aligned correctly before subtraction to obtain the resulting GP.
• Valid GP sum condition: The formula for the sum of an infinite GP, $S_{\infty} = \frac{a}{1-r}$, is only applicable when the absolute value of the common ratio $|r|$ is strictly less than $1$. In this case, $r=1/2$, so the condition is satisfied.

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Summary

The problem involves an infinite product that can be simplified by expressing all terms with a common base, $2$. This transforms the product into $2$ raised to the power of an infinite sum. The infinite sum is identified as an Arithmetico-Geometric Progression (AGP). By multiplying the AGP by its common ratio ($1/2$) and subtracting, we convert it into a standard infinite Geometric Progression (GP). The sum of this GP is calculated using the formula $S_{\infty} = \frac{a}{1-r}$, yielding a sum of $1$. Finally, substituting this sum back into the exponent of $2$ gives the value of the product as $2^1 = 2$.

The final answer is \boxed{2}.