a, b, c, d and e are integers .If a, b, c, d and e are geometric progression and lcm(m, n) is the least common multiple of m and n, then the maximum value of
`(1)/(1cm (a,b)) + (1)/(1cm (b,c)) + (1)/(1cm(c,d)) + (1)/(1cm (d,e)) ` is

Options

A. 1
B. `15/16`
C. `79/81`
D. `7/8`

Answer

To solve the problem, we need to find the maximum value of the expression:

\[ \frac{1}{\text{lcm}(a,b)} + \frac{1}{\text{lcm}(b,c)} + \frac{1}{\text{lcm}(c,d)} + \frac{1}{\text{lcm}(d,e)} \]

given that \(a, b, c, d, e\) are in geometric progression.

### Step 1: Define the terms in the geometric progression
Let’s assume:
- \(a = 1\) (the first term)
- \(r\) = common ratio

Then the terms can be expressed as:
- \(b = ar = 1 \cdot r = r\)
- \(c = ar^2 = 1 \cdot r^2 = r^2\)
- \(d = ar^3 = 1 \cdot r^3 = r^3\)
- \(e = ar^4 = 1 \cdot r^4 = r^4\)

So, we have:
- \(a = 1\)
- \(b = r\)
- \(c = r^2\)
- \(d = r^3\)
- \(e = r^4\)

### Step 2: Calculate the LCM for each pair
Now we calculate the LCM for each pair:
1. \(\text{lcm}(a, b) = \text{lcm}(1, r) = r\)
2. \(\text{lcm}(b, c) = \text{lcm}(r, r^2) = r^2\)
3. \(\text{lcm}(c, d) = \text{lcm}(r^2, r^3) = r^3\)
4. \(\text{lcm}(d, e) = \text{lcm}(r^3, r^4) = r^4\)

### Step 3: Substitute LCM values into the expression
Now substituting these into the expression, we get:
\[ \frac{1}{\text{lcm}(a,b)} + \frac{1}{\text{lcm}(b,c)} + \frac{1}{\text{lcm}(c,d)} + \frac{1}{\text{lcm}(d,e)} = \frac{1}{r} + \frac{1}{r^2} + \frac{1}{r^3} + \frac{1}{r^4} \]

### Step 4: Simplify the expression
To simplify this, we can factor out \(\frac{1}{r}\):
\[ = \frac{1}{r} \left( 1 + \frac{1}{r} + \frac{1}{r^2} + \frac{1}{r^3} \right) \]

The expression inside the parentheses is a geometric series with first term \(1\) and common ratio \(\frac{1}{r}\):
\[ 1 + \frac{1}{r} + \frac{1}{r^2} + \frac{1}{r^3} = \frac{1 - \left(\frac{1}{r}\right)^4}{1 - \frac{1}{r}} = \frac{1 - \frac{1}{r^4}}{1 - \frac{1}{r}} = \frac{r^4 - 1}{r^4 - r^3} \]

### Step 5: Combine and find the maximum value
Thus, the entire expression becomes:
\[ \frac{1}{r} \cdot \frac{r^4 - 1}{r^4 - r^3} = \frac{r^4 - 1}{r(r^4 - r^3)} = \frac{r^4 - 1}{r^4 - r^3} \]

To maximize this expression, we can set \(r = 2\):
\[ = \frac{2^4 - 1}{2^4 - 2^3} = \frac{16 - 1}{16 - 8} = \frac{15}{8} \]

### Conclusion
The maximum value of the expression is:
\[ \frac{15}{16} \]