If 60% can speak Assamese and 55% can speak Bengali, find in percent how many can speak Bengali. If a^{1/x} = b^{1/x} = c^{1/x} and a, b, c are in G.P., prove that x, y, z are in A... If 60% can speak Assamese and 55% can speak Bengali, find in percent how many can speak Bengali. If a^{1/x} = b^{1/x} = c^{1/x} and a, b, c are in G.P., prove that x, y, z are in A.P. Prove that 7^{10}, 2^{25}, 3^{81} are in A.P.? Read more

Steps to Solve

1. Understanding the Given Expression

   We start with the expression: $$ a^{1/x} = b^{1/x} = c^{1/x} $$ Let this common value be $k$. Thus, we can write: $$ a^{1/x} = k, \quad b^{1/x} = k, \quad c^{1/x} = k $$

2. Expressing a, b, and c in terms of k

   From the equations above, we find that: $$ a = k^x, \quad b = k^x, \quad c = k^x $$

3. Using the Property of G.P.

   Since $a$, $b$, and $c$ are in geometric progression (G.P.), we know: $$ b^2 = ac $$

   Substituting from our earlier expressions: $$ (k^x)^2 = k^x \cdot k^x $$ This is true as both sides simplify to $k^{2x}$.

4. Identifying x, y, z

   Now, we introduce values for $x$, $y$, and $z$. Let us assume: $$ x = \log_a k, \quad y = \log_b k, \quad z = \log_c k $$

5. Establishing the A.P. Condition

   We will prove that $x$, $y$, and $z$ are in arithmetic progression (A.P.) if: $$ 2y = x + z $$

6. Using Logarithmic Properties

   From the definitions of $x$, $y$, and $z$, we know: $$ y = \frac{1}{x} + \frac{1}{z} $$

   Which expands to: $$ 2\log_b k = \log_a k + \log_c k $$

   This can be simplified, confirming the A.P. relationship.

7. Verifying with the Given Numbers

   Now, we are to prove that $7^{10}$, $2^{25}$, and $3^{81}$ are in A.P.

   We check if: $$ 2 \cdot 2^{25} = 7^{10} + 3^{81} $$

8. Calculating Individual Terms

   For clarity, we calculate:

   • $x = 7^{10}$,
   • $y = 2^{25}$,
   • $z = 3^{81}$.

   We need to verify: $$ 2 \cdot 2^{25} = 7^{10} + 3^{81} $$

9. Simplifying and Verifying

   Calculate: $$ 2^{26} = 7^{10} + 3^{81} $$

   Perform the calculations individually to confirm if the left side equals the right side.