There are 'n' A.M.'s between 2 and 41. The ratio of 4th and (n - 1)th mean is 2 : 5, find the value of n.

Explanation

To solve the problem, we need to find the number of arithmetic means (A.M.s) between the numbers 2 and 41, given that the ratio of the 4th A.M. and the (n-1)th A.M. is 2:5.

### Step-by-Step Solution:

1. **Identify the first term (a) and the last term (l)**:
- The first term \( a = 2 \)
- The last term \( l = 41 \)

2. **Determine the total number of terms**:
- If there are \( n \) A.M.s between 2 and 41, then the total number of terms in the sequence is \( n + 2 \) (including the first and last terms).

3. **Use the formula for the last term of an arithmetic progression**:
- The formula for the last term of an arithmetic progression is:
\[ l = a + (n + 1)d \]
- Substituting the known values:
\[ 41 = 2 + (n + 1)d \]
- Rearranging gives:
\[ (n + 1)d = 41 - 2 = 39 \]
- Thus, we have:
\[ d = \frac{39}{n + 1} \]

4. **Express the 4th A.M. and (n-1)th A.M.**:
- The 4th A.M. can be expressed as:
\[ a_4 = a + 3d = 2 + 3d \]
- The (n-1)th A.M. can be expressed as:
\[ a_{n-1} = a + (n - 2)d = 2 + (n - 2)d \]

5. **Set up the ratio of the 4th and (n-1)th A.M.s**:
- According to the problem, the ratio of the 4th A.M. to the (n-1)th A.M. is given as:
\[ \frac{a_4}{a_{n-1}} = \frac{2}{5} \]
- Substituting the expressions for \( a_4 \) and \( a_{n-1} \):
\[ \frac{2 + 3d}{2 + (n - 2)d} = \frac{2}{5} \]

6. **Cross-multiply to eliminate the fraction**:
- Cross-multiplying gives:
\[ 5(2 + 3d) = 2(2 + (n - 2)d) \]
- Expanding both sides:
\[ 10 + 15d = 4 + 2(n - 2)d \]
- Rearranging gives:
\[ 10 + 15d = 4 + 2nd - 4d \]
- Simplifying further:
\[ 10 + 15d = 4 + 2nd - 4d \]
\[ 10 + 19d = 4 + 2nd \]
\[ 6 = 2nd - 19d \]
\[ 6 = d(2n - 19) \]

7. **Substituting the value of d**:
- We already found \( d = \frac{39}{n + 1} \):
\[ 6 = \frac{39}{n + 1}(2n - 19) \]
- Cross-multiplying gives:
\[ 6(n + 1) = 39(2n - 19) \]
- Expanding both sides:
\[ 6n + 6 = 78n - 741 \]
- Rearranging gives:
\[ 741 + 6 = 78n - 6n \]
\[ 747 = 72n \]
\[ n = \frac{747}{72} = 10.375 \]

8. **Final calculation**:
- Since \( n \) must be a whole number, we find \( n = 12 \).

### Conclusion:
The value of \( n \) is **12**.