What is the value of 13 + 14 ……. 20?

Options

A. 182
B. 144
C. 132
D. 124

Answer

To find the value of the series \( 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 \), we can use two methods: direct addition and the formula for the sum of an arithmetic progression (AP).

### Method 1: Direct Addition

1. **List the numbers**: Write down the numbers from 13 to 20.
\[ 13, 14, 15, 16, 17, 18, 19, 20 \]

2. **Add the numbers**:
\[ 13 + 14 = 27 \]
\[ 27 + 15 = 42 \]
\[ 42 + 16 = 58 \]
\[ 58 + 17 = 75 \]
\[ 75 + 18 = 93 \]
\[ 93 + 19 = 112 \]
\[ 112 + 20 = 132 \]

So, the total sum is \( 132 \).

### Method 2: Using the Arithmetic Progression Formula

1. **Identify the first term (a)**: The first term \( a = 13 \).

2. **Identify the last term (l)**: The last term \( l = 20 \).

3. **Identify the number of terms (n)**: Count the terms from 13 to 20.
- The terms are: 13, 14, 15, 16, 17, 18, 19, 20.
- There are 8 terms.

4. **Identify the common difference (d)**: The common difference \( d = 1 \) (since each term increases by 1).

5. **Use the formula for the sum of an AP**:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting the values:
\[ S_8 = \frac{8}{2} \times (2 \times 13 + (8 - 1) \times 1) \]
\[ S_8 = 4 \times (26 + 7) \]
\[ S_8 = 4 \times 33 = 132 \]

Thus, the sum of the series is \( 132 \).

### Final Answer
The value of \( 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 \) is \( 132 \).

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