Question

If for $x, y \in \mathbb{R}, x>0, y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \dots$ upto $\infty$ terms and $\frac{2 + 4 + 6 + \dots + 2y}{3 + 6 + 9 + \dots + 3y} = \frac{4}{\log_{10} x}$, then the ordered pair $(x, y)$ is equal to :

• A. $(10^6, 6)$
• B. $(10^2, 3)$
• C. $(10^4, 6)$
• D. $(10^6, 9)$

Hint:

In the fraction $\frac{\sum 2i}{\sum 3i}$, the variables cancel out to give exactly $2/3$ regardless of $y$. This quickly simplifies the second equation.

Answer 1

The Correct Option is D

To solve the given problem, we need to determine the ordered pair (x, y) given the equation:

\frac{2 + 4 + 6 + \dots + 2y}{3 + 6 + 9 + \dots + 3y} = \frac{4}{\log_{10} x}.

Let's address the elements of this calculation step-by-step:

1. Find the sum of the series for y:
   

   We have an infinite series: y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \dots

   This series is a geometric progression with the first term a = \log_{10} x and the common ratio r = \frac{1}{3}.

   The sum of an infinite geometric series is given by S = \frac{a}{1 - r}.

   Thus, the sum y is:

   y = \frac{\log_{10} x}{1 - \frac{1}{3}} = \frac{\log_{10} x}{\frac{2}{3}} = \frac{3}{2} \log_{10} x.

2. Compute the sums in the given fraction:
   

   The numerator is a sum of an arithmetic series: 2 + 4 + 6 + \dots + 2y with first term 2 and common difference 2. If there are n terms, the sum is given by:

   S = \frac{n}{2}(2 \cdot 2 + (n-1) \cdot 2) = n(n+1).

   In this case, n = y, so the sum becomes y(y + 1).

   The denominator is similar: 3 + 6 + 9 + \dots + 3y, yielding:

   D = \frac{y}{2}(2 \cdot 3 + (y-1) \cdot 3) = \frac{3y(y + 1)}{2}.

3. Substitute into the given equation:
   

   Substituting these into the equation:

   \frac{y(y + 1)}{\frac{3y(y + 1)}{2}} = \frac{4}{\log_{10} x}

   Simplifying gives:

   2/3 = \frac{4}{\log_{10} x}

   Solving for x:

   \log_{10} x = 6, so x = 10^6.

4. Find y:
   

   From y = \frac{3}{2} \log_{10} x and we found \log_{10} x = 6:

   y = \frac{3}{2} \times 6 = 9.

Thus, the correct ordered pair (x, y) is (10^6, 9).