If the third and 11th terms of an A.P. are 8 and 20 respectively, find the sum of first ten terms.

Options

A. `105(1)/(2)`
B. 108
C. `117(1)/(2)`
D. `203(1)/(2)`

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.).

### Step 1: Understand the given information
We know that:
- The 3rd term \( T_3 = 8 \)
- The 11th term \( T_{11} = 20 \)

### Step 2: Write the formulas for the terms of an A.P.
The general formula for the \( n \)-th term of an A.P. is given by:
\[ T_n = A + (n - 1)D \]
where \( A \) is the first term and \( D \) is the common difference.

### Step 3: Set up equations using the given terms
From the information provided:
1. For the 3rd term:
\[ T_3 = A + (3 - 1)D = A + 2D = 8 \]
This is our **Equation 1**.

2. For the 11th term:
\[ T_{11} = A + (11 - 1)D = A + 10D = 20 \]
This is our **Equation 2**.

### Step 4: Solve the equations
We have the two equations:
1. \( A + 2D = 8 \) (Equation 1)
2. \( A + 10D = 20 \) (Equation 2)

Now, we can eliminate \( A \) by subtracting Equation 1 from Equation 2:
\[ (A + 10D) - (A + 2D) = 20 - 8 \]
This simplifies to:
\[ 10D - 2D = 12 \]
\[ 8D = 12 \]
Now, solve for \( D \):
\[ D = \frac{12}{8} = \frac{3}{2} \]

### Step 5: Find the value of \( A \)
Now that we have \( D \), we can substitute it back into Equation 1 to find \( A \):
\[ A + 2D = 8 \]
Substituting \( D = \frac{3}{2} \):
\[ A + 2 \times \frac{3}{2} = 8 \]
\[ A + 3 = 8 \]
\[ A = 8 - 3 = 5 \]

### Step 6: Calculate the sum of the first 10 terms
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \left( 2A + (n - 1)D \right) \]
For \( n = 10 \):
\[ S_{10} = \frac{10}{2} \left( 2 \times 5 + (10 - 1) \times \frac{3}{2} \right) \]
\[ = 5 \left( 10 + 9 \times \frac{3}{2} \right) \]
\[ = 5 \left( 10 + \frac{27}{2} \right) \]
Convert \( 10 \) into a fraction with a denominator of 2:
\[ 10 = \frac{20}{2} \]
So,
\[ S_{10} = 5 \left( \frac{20}{2} + \frac{27}{2} \right) \]
\[ = 5 \left( \frac{47}{2} \right) \]
\[ = \frac{235}{2} \]

### Final Answer
Thus, the sum of the first 10 terms is:
\[ S_{10} = \frac{235}{2} \text{ or } 117.5 \]