`a_1,a_2,a_3` ,…., `a_n` from an A.P. Then the sum `sum_(i=1)^10 (a_i a_(i+1)a_(i+2))/(a_i + a_(i+2))` where `a_1=1` and `a_2=2` is :

Options

A. `495/3`
B. `495/4`
C. `495/2`
D. `495`

Answer

To solve the problem, we need to find the sum

\[ S = \sum_{i=1}^{10} \frac{a_i a_{i+1} a_{i+2}}{a_i + a_{i+2}} \]

where \(a_1 = 1\) and \(a_2 = 2\) are the first two terms of an arithmetic progression (A.P.).

### Step 1: Determine the general term of the A.P.
Since \(a_1 = 1\) and \(a_2 = 2\), we can find the common difference \(d\):

\[ d = a_2 - a_1 = 2 - 1 = 1 \]

Thus, the general term of the A.P. can be expressed as:

\[ a_n = a_1 + (n-1)d = 1 + (n-1) \cdot 1 = n \]

### Step 2: Write out the terms for \(a_i\), \(a_{i+1}\), and \(a_{i+2}\)
From the general term, we have:

- \(a_i = i\)
- \(a_{i+1} = i + 1\)
- \(a_{i+2} = i + 2\)

### Step 3: Substitute these terms into the sum
Now we substitute these into the sum:

\[ S = \sum_{i=1}^{10} \frac{i(i+1)(i+2)}{i + (i+2)} \]

### Step 4: Simplify the denominator
The denominator simplifies as follows:

\[ i + (i + 2) = 2i + 2 = 2(i + 1) \]

### Step 5: Rewrite the sum
Now we can rewrite the sum:

\[ S = \sum_{i=1}^{10} \frac{i(i+1)(i+2)}{2(i+1)} = \sum_{i=1}^{10} \frac{i(i+2)}{2} \]

### Step 6: Factor out the constant
We can factor out the constant \( \frac{1}{2} \):

\[ S = \frac{1}{2} \sum_{i=1}^{10} i(i+2) \]

### Step 7: Expand the summation
Now we expand \(i(i+2)\):

\[ i(i+2) = i^2 + 2i \]

Thus,

\[ S = \frac{1}{2} \left( \sum_{i=1}^{10} i^2 + 2\sum_{i=1}^{10} i \right) \]

### Step 8: Use summation formulas
We use the formulas for the sums:

1. \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\)
2. \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\)

For \(n = 10\):

\[ \sum_{i=1}^{10} i = \frac{10 \cdot 11}{2} = 55 \]

\[ \sum_{i=1}^{10} i^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385 \]

### Step 9: Substitute back into the equation
Now we substitute these values back into the equation:

\[ S = \frac{1}{2} \left( 385 + 2 \cdot 55 \right) = \frac{1}{2} \left( 385 + 110 \right) = \frac{1}{2} \cdot 495 = \frac{495}{2} \]

### Final Answer
Thus, the final answer is:

\[ S = 247.5 \]