If the sum of n terms of three A.P.'s are `S_1, S_2 and S_3` The first term of each A.P. is unity and the common differences are 1, 2 and 3 respectively, then `(S_1+S_3)/S_2` is equal to

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \((S_1 + S_3) / S_2\) where \(S_1\), \(S_2\), and \(S_3\) are the sums of the first \(n\) terms of three different arithmetic progressions (APs) with the same first term of 1 and different common differences.

### Step 1: Write the formula for the sum of the first \(n\) terms of an AP
The formula for the sum of the first \(n\) terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

### Step 2: Calculate \(S_1\)
For the first AP:
- First term \(a = 1\)
- Common difference \(d_1 = 1\)

Using the formula:
\[ S_1 = \frac{n}{2} \times (2 \times 1 + (n - 1) \times 1) = \frac{n}{2} \times (2 + n - 1) = \frac{n}{2} \times (n + 1) = \frac{n(n + 1)}{2} \]

### Step 3: Calculate \(S_2\)
For the second AP:
- First term \(a = 1\)
- Common difference \(d_2 = 2\)

Using the formula:
\[ S_2 = \frac{n}{2} \times (2 \times 1 + (n - 1) \times 2) = \frac{n}{2} \times (2 + 2(n - 1)) = \frac{n}{2} \times (2 + 2n - 2) = \frac{n}{2} \times 2n = n^2 \]

### Step 4: Calculate \(S_3\)
For the third AP:
- First term \(a = 1\)
- Common difference \(d_3 = 3\)

Using the formula:
\[ S_3 = \frac{n}{2} \times (2 \times 1 + (n - 1) \times 3) = \frac{n}{2} \times (2 + 3(n - 1)) = \frac{n}{2} \times (2 + 3n - 3) = \frac{n}{2} \times (3n - 1) = \frac{n(3n - 1)}{2} \]

### Step 5: Calculate \((S_1 + S_3)\)
Now we can find \(S_1 + S_3\):
\[ S_1 + S_3 = \frac{n(n + 1)}{2} + \frac{n(3n - 1)}{2} = \frac{n(n + 1 + 3n - 1)}{2} = \frac{n(4n)}{2} = 2n^2 \]

### Step 6: Calculate \(\frac{S_1 + S_3}{S_2}\)
Now, we can find \(\frac{S_1 + S_3}{S_2}\):
\[ \frac{S_1 + S_3}{S_2} = \frac{2n^2}{n^2} = 2 \]

### Final Answer
Thus, the value of \(\frac{S_1 + S_3}{S_2}\) is \(2\).