The third term of an AP is 1/5 and the 5th term is 1/3 find the sum of 15 terms of the AP:

Options

A. `(1/15)`
B. `(3/5)`
C. 8
D. 15

Answer

To solve the problem step by step, we will follow the information provided in the question regarding the arithmetic progression (AP).

### Step 1: Understand the given information
We know that:
- The 3rd term of the AP is \( \frac{1}{5} \).
- The 5th term of the AP is \( \frac{1}{3} \).

### Step 2: Write the formulas for the terms
The nth term of an AP can be expressed as:
\[ T_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

For the 3rd term:
\[ T_3 = a + 2d = \frac{1}{5} \quad \text{(1)} \]

For the 5th term:
\[ T_5 = a + 4d = \frac{1}{3} \quad \text{(2)} \]

### Step 3: Set up the equations
From the equations (1) and (2), we have:
1. \( a + 2d = \frac{1}{5} \)
2. \( a + 4d = \frac{1}{3} \)

### Step 4: Subtract the first equation from the second
Subtract equation (1) from equation (2):
\[ (a + 4d) - (a + 2d) = \frac{1}{3} - \frac{1}{5} \]
This simplifies to:
\[ 2d = \frac{1}{3} - \frac{1}{5} \]

### Step 5: Find a common denominator and solve for \( d \)
The common denominator for 3 and 5 is 15. Thus:
\[ \frac{1}{3} = \frac{5}{15}, \quad \frac{1}{5} = \frac{3}{15} \]
So:
\[ 2d = \frac{5}{15} - \frac{3}{15} = \frac{2}{15} \]
Now divide both sides by 2:
\[ d = \frac{2}{15} \times \frac{1}{2} = \frac{1}{15} \]

### Step 6: Substitute \( d \) back to find \( a \)
Now we can substitute \( d \) back into equation (1):
\[ a + 2\left(\frac{1}{15}\right) = \frac{1}{5} \]
This simplifies to:
\[ a + \frac{2}{15} = \frac{3}{15} \]
Thus:
\[ a = \frac{3}{15} - \frac{2}{15} = \frac{1}{15} \]

### Step 7: Find the sum of the first 15 terms
The sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
For \( n = 15 \):
\[ S_{15} = \frac{15}{2} \times \left(2 \times \frac{1}{15} + (15-1) \times \frac{1}{15}\right) \]
This simplifies to:
\[ S_{15} = \frac{15}{2} \times \left(\frac{2}{15} + \frac{14}{15}\right) = \frac{15}{2} \times \frac{16}{15} \]
Now, simplifying:
\[ S_{15} = \frac{15 \times 16}{2 \times 15} = \frac{16}{2} = 8 \]

### Final Answer
The sum of the first 15 terms of the AP is \( 8 \).

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