Let u,v,w be rela numbers in geometric progression such that `u gt v gt w.` Suppose `u ^(40)= v ^(n) = w ^(60).` Find the value of n .

Answer

To solve the problem, we start with the given conditions. Let \( u, v, w \) be real numbers in geometric progression such that \( u > v > w \). We are given that:

\[ u^{40} = v^n = w^{60} \]

### Step 1: Express \( v \) in terms of \( u \) and \( w \)

Since \( u, v, w \) are in geometric progression, we know that:

\[ v^2 = u \cdot w \]

Thus, we can express \( v \) as:

\[ v = \sqrt{u \cdot w} \]

### Step 2: Relate \( u \) and \( w \)

From the equations \( u^{40} = w^{60} \), we can express \( u \) in terms of \( w \):

\[ u^{40} = w^{60} \]

Taking the 40th root on both sides, we get:

\[ u = w^{\frac{60}{40}} = w^{\frac{3}{2}} \]

### Step 3: Substitute \( u \) into the expression for \( v \)

Now substitute \( u = w^{\frac{3}{2}} \) into the expression for \( v \):

\[ v = \sqrt{u \cdot w} = \sqrt{w^{\frac{3}{2}} \cdot w} = \sqrt{w^{\frac{3}{2} + 1}} = \sqrt{w^{\frac{5}{2}}} = w^{\frac{5}{4}} \]

### Step 4: Set up the equation for \( v^n \)

Now we have:

\[ v^n = (w^{\frac{5}{4}})^n = w^{\frac{5n}{4}} \]

### Step 5: Equate \( v^n \) and \( w^{60} \)

Since \( v^n = w^{60} \), we can equate the exponents:

\[ w^{\frac{5n}{4}} = w^{60} \]

This implies:

\[ \frac{5n}{4} = 60 \]

### Step 6: Solve for \( n \)

Now, we can solve for \( n \):

\[ 5n = 60 \cdot 4 \]
\[ 5n = 240 \]
\[ n = \frac{240}{5} = 48 \]

### Conclusion

Thus, the value of \( n \) is:

\[ \boxed{48} \]