If `2lamda` is the number of ways of selecting 3 member subset of {1,2,3, . .,29}, so that the number form of a GP with integer common ratio, then find the value of `lamda`

Explanation

To solve the problem, we need to find the number of ways to select a 3-member subset from the set {1, 2, 3, ..., 29} such that the numbers form a geometric progression (GP) with an integer common ratio.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to select three numbers \( a, b, c \) from the set {1, 2, ..., 29} such that they form a GP. For three numbers to be in GP, they must satisfy the condition \( b^2 = ac \), where \( b \) is the middle term.

2. **Setting Up the GP**:
Let the first term be \( a \) and the common ratio be \( r \). Then the three terms can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)

For \( a, ar, ar^2 \) to be in the set {1, 2, ..., 29}, we need:
- \( a \) must be a positive integer.
- \( ar \) and \( ar^2 \) must also be less than or equal to 29.

3. **Finding Possible Values of \( a \) and \( r \)**:
We can express the conditions as:
- \( ar^2 \leq 29 \)
- This implies \( a \leq \frac{29}{r^2} \)

For each integer value of \( r \), we can find the corresponding values of \( a \).

4. **Calculating the Number of Valid Triples**:
We will consider different integer values of \( r \):
- For \( r = 1 \): The terms are \( a, a, a \) (only valid if \( a \leq 29 \)). There are 29 choices.
- For \( r = 2 \): The terms are \( a, 2a, 4a \). We need \( 4a \leq 29 \) which gives \( a \leq 7 \). So there are 7 choices.
- For \( r = 3 \): The terms are \( a, 3a, 9a \). We need \( 9a \leq 29 \) which gives \( a \leq 3 \). So there are 3 choices.
- For \( r = 4 \): The terms are \( a, 4a, 16a \). We need \( 16a \leq 29 \) which gives \( a \leq 1 \). So there is 1 choice.
- For \( r \geq 5 \): The terms will exceed 29 for any positive \( a \).

5. **Summing Up the Choices**:
Now we sum the valid choices:
- For \( r = 1 \): 29 choices
- For \( r = 2 \): 7 choices
- For \( r = 3 \): 3 choices
- For \( r = 4 \): 1 choice

Total choices = \( 29 + 7 + 3 + 1 = 40 \).

6. **Finding \( \lambda \)**:
According to the problem, \( 2\lambda = 40 \). Therefore, we can find \( \lambda \):
\[ \lambda = \frac{40}{2} = 20 \]

### Final Answer:
Thus, the value of \( \lambda \) is \( 20 \).

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